Hunter Biden’s name is ringing throughout the political universe — from the halls of Congress to the annals of social media and the cornfields of Iowa, messing with former Vice President Joseph R. Biden’s “No Malarkey” mojo in the Democratic presidential race.

The 49-year-old has emerged as a central character in the impeachment inquiry on Capitol Hill and is complicating life for Mr. Biden on the campaign trail. Questions about Hunter’s tabloid-worthy personal life, as well as his cushy position on the board of a Ukranian energy company, are muddying his father’s message.

“It is frustrating because it is the kind of thing that people know is going to get under his skin,” said Jan Bauer, a member of the Democratic National Committee from Iowa. “It is going to irritate him, so they are going to do it more and more.”

It is part of the no-holds-barred scrutiny that comes with running for the highest office in the land. The candidates’ resumes and rhetoric are put under a microscope, as are their friends and family.

President Trump’s attempts to tout his business acumen in 2016 faced pushback from Democrats who demanded to see his tax returns.

Hillary Clinton tried to run on her governing experience and intellect but struggled to get out from under questions about her secret email server.

Eight years earlier, then-Sen. Barack Obama ran as a unifier and faced pushback over some of the incendiary remarks by his longtime pastor, the Rev. Jeremiah Wright, and in 2004 then-Sen. John F. Kerry’s attempts to play up his military experience were undercut by the Swift Boat Veterans for Truth ads.

“You always attack your opponent’s perceived strength, and once you take their strength off the table or you nullify it, it makes it much more of an event campaign, if you will, and it is even better when it happens to be true,” said Chris LaCivita, a national Republican Party consultant who served as chief strategist for the Swift Boat Veterans.

Mr. LaCivita said Mr. Biden is trying to downplay the attacks in much the same way Mr. Kerry did.

“By refusing to acknowledge, refusing to engage on what his son knew, what he actually did, by ignoring it and not addressing it, he is putting himself in the same position as John Kerry did in 2004,” he said. “But you know old habits from long-established politicians like John Kerry and Joe Biden are hard to shake.”

As fate would have it, Mr. Kerry, who endorsed Mr. Biden last week, joined him in New Hampshire over the weekend and on his “No Malarkey” bus tour through Iowa, where Mr. Biden said he didn’t know what Hunter Biden did to earn $50,000 per month on the board of Burisma Holdings while his father was vice president.

Asked if it’s something he wants to get to the bottom of, Mr. Biden told Axios: “No. Because I trust my son.”

“There’s nothing on its face that was wrong,” he said. “Look, if you want to talk about problems, let’s talk about Trump’s family. I mean, come on.

Pressed further, he said: “You know, there’s not one single bit of evidence, not one little tiny bit to say anything done was wrong … but you keep asking me these questions.

“But I’m not worried about it. Look, the American public knows me,” he said.

The campaign swing also featured a headline-grabbing showdown with an 83-year-old man who accused Mr. Biden of “selling access to the president” and landing Hunter, who had no experience in the energy business.

“You’re a damn liar, man. That’s not true,” Mr. Biden snapped.

A little over a week earlier, Mr. Biden had another tense exchange, this time with a reporter who asked him to comment on a woman’s claims that a paternity test proves that Hunter is the father of her baby.

“No, that’s a private matter and I have no comment,” Mr. Biden said. “Only you would ask that. You’re a good man. You’re a good man. Classy.”

It was the latest in a series of ugly headlines about Hunter Biden, who also had a crack cocaine habit and a romantic relationship with his brother’s widow.

Hunter Biden’s name came up repeatedly during the impeachment hearings. Republican lawmakers peppered witnesses with questions about the legitimacy of requesting an investigation of Mr. Biden’s son.

Several officials acknowledged that Hunter Biden’s business dealings in Ukraine were suspect and posed a potential conflict of interest for his father.

Hunter Biden’s name came up at least eight times last week when the House Judiciary Committee heard from a panel of legal scholars about whether the case against Mr. Trump warranted impeachment.

“Today we are undertaking a largely academic exercise instead of hearing from fact witnesses like … Hunter Biden,” said Rep. Steven J. Chabot, an Ohio Republican on the committee.

He was among several Republican lawmakers who have demanded testimony from Hunter Biden.

Senate Republicans are talking about calling Hunter Biden and his father to testify if impeachment reaches the upper chamber.

Mr. Trump’s request of Ukrainian President Volodymyr Zelensky for a “favor” in investigating Hunter Biden is what Democrats describe as an attempt to “dig up dirt” to help his reelection campaign. They say that request constitutes abusing Oval Office power for personal gain.

Mr. Trump has acknowledged that he wanted an investigation into alleged corruption involving Mr. Biden and his son Hunter, who landed the post at Burisma in 2014. At the time, his father was the point man for Obama White House policy in the country, which is notorious for corruption, especially in the energy industry.

The former vice president recently boasted of forcing Ukrainian leaders to fire the country’s chief prosecutor in spring 2016 by threatening to block a $1 billion U.S. loan guarantee. The prosecutor was widely viewed as not doing enough to combat corruption. But the prosecutor, Viktor Shokin, had looked into corruption allegations against Burisma and Mykola Zlochevsky, the Ukrainian oligarch running the company.

“Even if Hunter Biden engaged in no crimes regarding his sitting on the board of Burisma, if an investigation led to the bankruptcy of the corrupt company, Hunter Biden’s lucrative position on the Burisma board would have been eliminated along with his $50,000 a month payments. That was his stake in a potential prosecution involving the company,” Paul Taylor, the Republican chief counsel for the House Judiciary Committee, said in questioning the legal scholars.

It has become a matter of debate in Washington about whether the Bidens should have to testify at a Senate impeachment trial. The former vice president said he won’t do it.

White House senior adviser Kellyanne Conway challenged the Democrats’ narrative that Mr. Biden should be off limits during his presidential campaign.

“Is Joe Biden insulated, is his son insulated somehow because his father is running for president?” she asked reporters. “I would think that if you were running for president the scrutiny should be higher, the transparency demand should be greater — not lower. And if they have nothing to hide, they should be happy to come and testify.”

• David Sherfinski contributed to this report.

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