In January 2019, Hunter Biden sent a text message to his daughter Naomi.

“I Hope you all can do what I did and pay for everything for this entire family Fro 30 years,” he wrote in the typo-filled message. “It’s really hard. But don’t worry unlike Pop I won’t make you give me half your salary.”

As they pursue their impeachment inquiry into President Biden, House Republicans have seized on that message and others sent to or from his son to try to link the president — “Pop,” in this case — to Hunter Biden’s business activities, suggesting that members of the Biden family reaped millions from a global influence-peddling scheme.

In the case of the message from Hunter Biden to his daughter, Republicans have portrayed it as evidence that he was privately acknowledging that he split his income with his father, who in early 2019 was out of office but preparing to begin the campaign that would put him in the White House. That income included the millions of dollars that Hunter Biden received from Burisma Holdings, a Ukrainian gas company, and from a company owned by a Chinese energy tycoon.

At a hearing in September, Representative Byron Donalds, Republican of Florida, displayed the message and asked a witness, Bruce G. Dubinsky, a forensic accountant, “If you saw a text message like this in a potential money laundering operation, or a potential pay-for-play operation, would you be looking for information related to money going from son to father?”

Mr. Dubinsky answered, “Absolutely.”

But a close examination of the circumstances surrounding the 2019 text message, along with others that have been cited by Republicans during the impeachment inquiry and elsewhere to suggest that Hunter Biden’s foreign income was shared with or benefited his father, shows the extent to which the contents of the communications have been misunderstood or outright distorted. And while it does not rule out the possibility that House Republicans could unearth evidence showing wrongdoing by President Biden, it underscores the flimsy nature of the material they have presented publicly so far.

The story behind the message, as explained by the Bidens and backed up by other interviews and a review of Hunter Biden’s emails and text messages, offers sometimes unflattering insights into the family’s finances and internal dynamics. And it adds more detail to what is known about Hunter Biden’s erratic and irresponsible behavior while in the throes of addiction.

But it is very different from the story being promoted by House Republicans and their allies.

Rather than evidence that Hunter had split his foreign income with his father, the 2019 message was a reference to a story from Hunter’s youth that he repeated to his daughters when they became teenagers. It was prompted by a family dispute, fueled by Hunter Biden’s drug use, money troubles and personal resentments, according to the review of Hunter Biden’s communications and interviews with Biden aides, family friends, and Hunter and Naomi Biden.

It started with a freak ski accident at the start of 2019.

Naomi Biden, 25 at the time, and her then-boyfriend, Peter Neal, were having lunch at the slope-side Handle Bar Restaurant & Pub at the Four Seasons Jackson Hole ski resort in Wyoming. They received a frantic phone call from Naomi’s younger sister Finnegan, who had skipped lunch with them to continue skiing.

Finnegan told Naomi that she had caught one of her skis on a branch and taken a hard fall, injuring her knee.

A rescue team from ski patrol loaded her into a toboggan to take her down the mountain.

“Please help,” Finnegan said, Naomi recalled.

Naomi immediately alerted her parents.

The accident occurred during one of the darkest periods in Hunter’s life. After the death of his brother, Beau, in 2015, he descended into a spiral of addiction and depression.

He and his wife at the time, Kathleen Buhle, separated, and their relationship deteriorated as his problems with addiction intensified, culminating in their acrimonious divorce in 2017.

During this period, Hunter was financially dependent on Burisma, the Ukrainian gas company, which appointed him to a board seat while his father was vice president and oversaw U.S. policy in Ukraine.

The Burisma money was substantial — a little under $500,000 in 2018, according to a recent federal indictment of Hunter on charges of tax violations, down from an initial $1 million a few years earlier — but it was not enough.

In addition to bankrolling his own lifestyle, which during this period often revolved around crack cocaine and chasing women, Hunter was responsible for covering most of his immediate family’s expenses, which included mortgages, alimony payments to Ms. Buhle, and tuition for their three girls to attend pricey private schools and college.

Finnegan’s skiing accident had fractured her tibia, and the injury prompted a heated tug of war between Hunter and Ms. Buhle over where Naomi should take Finnegan for medical treatment.

Ms. Buhle told Naomi to bring Finnegan to Washington, where she lived.

Hunter devised a different plan. At the time, he was in Newburyport, Mass., receiving ketamine infusion therapy, in hope of easing his addictions to crack and alcohol.

The treatment did not work, and when Hunter learned about Finnegan’s injury, he was desperate to show to everyone in the family that he could still be a responsible parent, family members said.

The doctor who had been treating Hunter with ketamine recommended an orthopedic surgeon at the Hospital for Special Surgery in New York. Hunter instructed Naomi to take Finnegan to New York and told her that he had even arranged for an ambulance to pick Finnegan up at the airport upon arrival to whisk her into surgery.

Instead of heeding her father, Naomi listened to her mother and took Finnegan to Washington.

Hunter felt dejected, and, while apparently under the influence of drugs, wrote a series of angry and often nonsensical messages to Naomi in which he threatened to cut her off financially.

“Find an apartment with Peter by next week,” Hunter instructed. “And send me the keys and leave all of my furniture and art. I love all of you. But I don’t receive any respect.”

Then he sent the text message that Republicans have used to suggest that Hunter’s foreign income was going to enrich his father.

The story behind Hunter’s allusion to handing half of his salary to his father can be traced back to his childhood, and to the family’s conflicted relationship with money and class.

The way Hunter tells it, he and his older brother, Beau, grew up in a middle-class family. But their father was drawn to grand estates, and they lived in a 10,000-square-foot mansion with its own ballroom in Greenville, Del. (To save money, Joe Biden did most of the home repairs and improvements himself, and in winter closed off large sections with drywall to save on heating costs.)

While the family occasionally went on ski trips to Vail, a winter playground for wealthier families, the Bidens made their own sandwiches so they would not have to buy the expensive lunches sold out on the mountain, recalled their host, Roger Harrison, a longtime family friend whose brother owned the condo where the Bidens stayed free of charge.

The elder Mr. Biden encouraged his sons to find weekend and summer jobs to help pay for their personal expenses. They mowed neighbors’ lawns. Hunter worked at the Brandywine Zoo, and they both got jobs at Delaware Cold Storage, where Hunter was responsible for unloading rail cars carrying frozen beef while Beau pushed papers in the front office.

“The boys had to pay for everything they had growing up,” said Brian McGlinchey, a high school classmate.

Their house was situated on a relatively large plot of land, and the elder Mr. Biden sold off some of the parcels to help pay for Beau to go to the University of Pennsylvania, in 1987.

The following year, Hunter started at Georgetown University, in Washington.

Ted Dziak, a chaplain-in-residence in Hunter’s freshman dorm, said, “He didn’t have the money that I would have thought that a U.S. senator’s son would have had, so he was always looking for funds, looking to find a job or doing things that would earn him a little bit of money for college.”

Hunter’s oft-told story about giving half of his salary to his father appeared to originate during his freshman year at Georgetown.

His roommate at the time recalled Hunter telling him and his twin brother “a million times” that then-Senator Biden encouraged him to work, saying, “You can keep half of the paycheck, but you have to hand over the other half for ‘room and board.’”

It was a story, and a theme, that Hunter continued to invoke, especially after he married Ms. Buhle and they had three daughters — Naomi, Finnegan and Maisy — all of whom attended Sidwell Friends, a costly Washington private school, where they were surrounded by wealthier families.

Hunter told close friends that he was worried that his daughters had become spoiled. According to family members, he would frequently tell them the story about how he had to work in college and pay half of his salary to his father, in hopes of encouraging them to be more self-sufficient.

When Naomi was in her senior year at the University of Pennsylvania, Hunter pressured her to take a job at a Greek restaurant.

“He said, ‘Now that you’re working, I’m taking away your allowance,’” she recalled in an interview. “And I just thought that was the craziest concept that I’d ever heard. So I’m doing this good thing and you’re taking away my allowance? I was so mad at him, so angry, and I specifically remember him in that instance saying, ‘When I was in college, I worked every single day, and I even had to give Pop half the money because he was paying for my college education.’”

Naomi continued: “I don’t necessarily believe that he gave my Pop half. It’s the classic parent saying, ‘You don’t know what it was like for me when I was growing up. I just had it so much harder than you.’ Throughout childhood, we would hear that a lot.”

When he repeated the message in the 2019 text to Naomi during the fight over her sister’s ski accident, she understood it both as an expression of Hunter’s addiction problems and a long-running family tale.

A few days after Hunter threatened to cut her off financially, Naomi sought to ease tensions with her father by sending him a message saying she missed him. “I’m sorry I disappointed you,” she wrote. “It makes me so sad.”

In another message, Naomi told her father that she and her sisters would be more responsible about spending money. Referring to herself, her sisters and their dealings with their grandfather, Joe Biden, Naomi wrote, “We’re all writing budgets to give pop and we feel really badly for all the additional stress we’ve caused.”

To help ease the strain on Hunter’s finances, Joe Biden agreed to cover tuition payments for the three girls.

When approached by The New York Times about Republican allegations regarding Hunter’s January 2019 message to her, Naomi said, she reread the whole exchange.

“It just brought me back,” she said. “I started crying. You really can’t make sense of these texts because they don’t make sense. It was such a hard time. These were his darkest days because of the drugs. It’s just, like, little bits of trauma and anger coming out.”

Naomi said it was immediately clear to her what Hunter meant when he referred to giving half of his salary to his father.

“Imagine if you read through anyone’s text messages,” she said. “Of course you can twist these things. And of course they’re not going to be understood by everybody, because there’s a lot of personal nuances and inferences.”

She continued: “He was repeating a story from his university days that I grew up hearing. Do people really think he was texting me things like, ‘I give pop half of the Burisma money’? No. That’s crazy.”

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