Republican presidential candidates, some of whom were stumping in the early-caucusing state of Iowa on Friday, largely derided the news that the prosecutor investigating President Biden’s son Hunter had been elevated to special counsel status.

Gov. Ron DeSantis of Florida, during a campaign stop in Audubon, Iowa, cast doubt on the independence of the special counsel, David C. Weiss, who had already been overseeing a yearslong investigation of the president’s son. “It just seems to me that they’re going to find a way to give him some type of soft-glove treatment,’’ he said.

And Nikki Haley, the former U.S. ambassador to the United Nations, spoke disdainfully of the new title and power for Mr. Weiss.

“I don’t think the American people trust the Department of Justice or anything this is going to do,” Ms. Haley said in an appearance on Fox News. “I think this was meant to be a distraction.”

At the same time, she called it a “response to the pressure that the Biden family is feeling” and called on House Republicans who have been investigating the Bidens “to keep their foot on the gas.” So far, the investigations have found no hard evidence that President Biden used his influence while vice president to benefit his son’s business deals.

Gov. Doug Burgum of North Dakota also said he doubted the independence of Mr. Weiss. In an interview while twirling around on the Iowa State Fair’s Ferris wheel, he called the move “too little, too late” and said few Republicans would view the step as a serious development, given Mr. Weiss’s role in offering Hunter Biden a plea deal in the case. That plea deal has fallen apart.

Mr. Weiss is a federal prosecutor in Delaware who was originally appointed by former President Donald J. Trump. He was left in his position by President Biden to continue the Hunter Biden inquiry to avoid the appearance that the president would seek special treatment for his son.

In a statement attributed to a spokesperson, Mr. Trump, who is being investigated by the special counsel Jack Smith, claimed without evidence that the Department of Justice has protected President Biden, Hunter Biden and other family members “for decades.” The statement cast doubt on Mr. Weiss’s independence and criticized him for not already bringing “proper charges after a four-year investigation” of Hunter Biden.

Mr. Smith has brought two indictments against Mr. Trump.

Not all of the candidates were disdainful of the appointment of the special counsel, which Republicans have urged for some time. Vivek Ramaswamy, who said last month that a special counsel was warranted, called the appointment of Mr. Weiss “good” on X, the site formerly known as Twitter. “Now let’s see if it’s more than a fig leaf,” he added.

Former Vice President Mike Pence, who was flipping cuts of pork at the Iowa State Fair on Friday, said he approved of the Department of Justice’s move to upgrade Mr. Weiss’s power.

“I think it’s about time that we saw the appointment of a special counsel to get to the bottom of not only what Hunter Biden was doing, but what the Biden family was doing,” Mr. Pence said. “The American people deserve answers, and I welcome the appointment.”

Anjali Huynh and Neil Vigdor contributed reporting from Iowa.

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