MARION, Iowa — Joseph R. Biden isn’t mentioning his son Hunter by name on the stump, but the former vice president is beating back allegations of his son’s corrupt deals in Ukraine as the 2020 Democratic presidential race heats up.

It is part of the strategy Mr. Biden is using to knock down his two biggest hurdles in the sprint to the nation’s leadoff caucuses next week: his age and his son.

On Monday, the 77-year-old boasted about his granite chin as he campaigned across eastern Iowa, urging voters to consider that he is still standing tall after being on the receiving end of a high-powered smear campaign from President Trump and his allies.

“They have spent over $12 [million] to $15 million telling lies about family, maligning my only surviving son,” Mr. Biden said during a campaign stop at the University of Northern Iowa in Cedar Falls. “As much as he has tried to destroy me and my family, I hope I have demonstrated I can take a punch.”

He upped the metaphorical ante at his second stop. “I’ve taken brickbats to the head,” he told a standing-room-only crowd that turned out to see him at a local banquet hall.

Hunter Biden’s lucrative business deals in Ukraine and China while his father was vice president have long raised eyebrows, though no action was taken during the Obama administration.

He also has kept his name in the tabloid newspapers with exploits such as getting kicked out of the Navy Reserves for cocaine use, doing stints in drug rehabilitation, and having a romantic relationship with his late brother’s widow.

The elder Mr. Biden does name-drop his late son, Beau, who died of brain cancer in 2015 after serving as attorney general of Delaware and reaching the rank of major in the Delaware Army National Guard.

Mr. Biden is within striking distance of winning the Iowa caucuses, according to polls that show many voters are still making up their minds.

Indeed, voters acknowledge they are twisting themselves in knots over whom to back next week.

They are weighing the pluses and minuses of the contenders and trying to game out which of them will be most palatable to their family members, neighbors and co-workers.

“We are trying to overthink what everybody else is doing,” said Allison, a retired John Deere employee who declined to give her last name. “Everyone is trying to outguess everyone else.”

The 65-year-old Democrat said voters are mulling questions such as: Are Sens. Elizabeth Warren and Bernard Sanders too liberal for most of the country to defeat Mr. Trump? Are voters willing to elect a woman or a gay man to the White House? How old is too old to be president, and how big of a drag will Hunter Biden be on his father?

The allegations of corruption involving the Bidens also hang over Mr. Trump’s impeachment trial in Washington.

House Democrats impeached Mr. Trump for abuse of power for allegedly withholding nearly $400 million in military aid to force Ukraine to investigate Mr. Biden, the point person in Ukraine for the Obama administration, and Hunter Biden, who had landed a high-paying position at Burisma Holdings, a Ukrainian energy company embroiled in unresolved corruption charges.

Democrats want to hear testimony from members of Mr. Trump’s inner circle, while Republicans counter they want to put the Bidens on the stand.

Mr. Biden has said that is not going to happen and dismisses the Republican line of attack altogether.

Instead, he says Mr. Trump is trying to divert attention away from his misdeeds and failures.

Mr. Biden said he is running to “restore the soul of the nation” and turn back the clock on the polarization that is tearing the nation apart.

“We have to heal this country,” he said Monday. “It is going to be hard as the devil, but the first thing we have to do is get rid of this ugly, ugly political atmosphere we have.”

Larry Dillinger, a retired union worker, said he is confident Mr. Biden can deliver on his promise to work across party lines.

“If we give up on that, we’ve got anarchy,” the 72-year-old said.

Meanwhile, voters generally say they trust that the Bidens didn’t do anything wrong in Ukraine and that Mr. Trump is trying to dirty the candidate in much the same fashion he did with political rival Hillary Clinton in 2016 in hopes of depressing Democratic turnout.

“I see no reason why they’d have to call him as a witness because it has nothing to do with Donald Trump and him trying to persuade this guy to mess with our elections,” Mr. Dillinger said.

“I am sure Donald Trump will bring it up and bring it up, and bring it up, and eventually some of that will hurt him,” he said. “But we will have his back.”

The Hunter Biden saga, though, is on the minds of some voters, including Candy Clough, who said she is disappointed with the “totally shady” behavior of the Bidens and eager to move past the sorts of scandals that plagued the Clinton and Trump administrations.

Ms. Clough, a 69-year-old music teacher, said she is “mad at [Mr. Biden] and his stupid son.”

“I think that is going to deep-six him,” she said. “I mean, what are you doing here? Did you think this through a little? Maybe it is nothing, but that kid has issues.”

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