MANCHESTER, N.H. — Ken French says he is open to backing former Vice President Joseph R. Biden in the first-in-the-nation presidential primary, but first he wants more clarity on whether his son Hunter’s business dealings in Ukraine were “on the up-and-up.”

“As a voter, that is a concern,” said Mr. French, 61. “Right now, there is a shadow cast over Joe Biden.”

The Bidens’ history in Ukraine is getting another look in the aftermath of a whistleblower complaint accusing President Trump of using threats to cut off U.S. military aid to pressure Ukrainian President Volodymyr Zelensky to investigate the Bidens and the son’s profitable business dealings in Ukraine.

Mr. Trump has launched a counteroffensive denying any quid pro quo. He said investigators would be better served looking into whether Mr. Biden, as vice president, sought the ouster of a Ukrainian prosecutor because he was looking into corruption at the country’s largest private gas company, Burisma, where Hunter Biden served on the board and reportedly raked in $50,000 per month.

Voters in New Hampshire are now trying to sort out who is telling the truth and whether the former vice president and his son were involved in shady business dealings as they consider their options in the 2020 primary race.

Mr. Biden maintains the prosecutor lost his job because he dropped the ball when it came to combating Ukraine’s “cancer of corruption.” Mr. Biden last year boasted that his threat to withhold $1 billion in aid led to the prosecutor’s ultimate dismissal.

He described Mr. Trump’s account as “bizarre.” He said the storyline has been debunked by every “credible journalist” and that it is a blatant attempt to divert attention away from what he called the president’s unhinged and dangerous behavior.

“It is such a blatant abuse of power, I don’t think it can stand,” Mr. Biden said this week on “Jimmy Kimmel Live,” where he moved a step closer to backing the impeachment push on Capitol Hill.

The dueling versions of events date back to 2014 when Mr. Biden became the Obama administration’s point person on Ukraine after the country’s president was ousted in a revolution. Weeks later, Hunter Biden landed his plum job on the board at Burisma.

The political dust-up has created an opening for some of Mr. Biden’s rivals in the 2020 presidential primary. Sen. Elizabeth Warren, who has sprinted past Mr. Biden in early primary state polls, has made ferreting out corruption the centerpiece of her campaign.

Ms. Warren used her opening this week when reporters asked whether she would allow the child of her vice president to serve on the board of a foreign company.

“No,” the Massachusetts Democrat told reporters at a campaign stop in Keene before saying, “I don’t know — I’d have to go back and look at the details.”

Her campaign later clarified to The Washington Post that ethics proposals Ms. Warren has introduced don’t prevent children of a vice president from serving on a company’s board.

Ford O’Connell, a Republican Party strategist with close ties to the White House, said the episode is playing in Ms. Warren’s favor.

“If you are Elizabeth Warren, you have to be giddy,” Mr. O’Connell said. “While defending him to the teeth, you’re hoping it sinks him.”

Other voters say Hunter Biden, who had no previous experience in the energy sector, probably benefited from his father’s political clout, but they don’t think he broke any laws and they don’t think it will dent Mr. Biden’s chances in the primary race.

Michael Clark said Mr. Biden has a “good rapport” with voters in New Hampshire.

“I would back him,” the 65-year-old said. “I like Joe Biden. I think he is a good man, and he is a fair guy, I believe.”

Mr. French, meanwhile, said he isn’t writing off Mr. Biden but will remain on the fence until he learns more — including how Hunter Biden landed his cushy job, what sort of role Mr. Biden played in securing the post and whether he did anything for the money he received.

“If there is an investigation, I want to see what they come up with,” he said.

• David Sherfinski contributed to this report.

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