But on the first day of the general election, Mr. DeSantis said on Fox News that “the last thing we need to do is monkey this up” — seen by many as a racist reference to Mr. Gillum, who is black — and he has been struggling to recover ever since. The gaffe was, in the words of one of his own allies, the political equivalent of throwing an interception on the opening play of the game.

The 40-year-old former congressman has consistently trailed in public polls ever since, and has only now started to narrow the gap in a state where governor’s races, not to mention presidential contests, are typically decided by the smallest of margins.

A range of Republican officials, from the White House to the Florida state house, believe Mr. DeSantis has squandered valuable time, partly because his own self-inflicted errors were overshadowing Mr. Gillum’s vulnerabilities, which include an F.B.I. investigation of government corruption in Tallahassee, an inquiry in which Mr. Gillum has said he is not a target.

Now with the election just over a month away, the DeSantis forces appear to be rebooting his campaign. Last week they announced the appointment of a new campaign manager, Susie Wiles, a veteran Republican operative who chaired Mr. Trump’s Florida team and was recruited by donors to beef up and bring order to Mr. DeSantis’s skeletal operation. And after leaving Florida briefly this week to raise money in Dallas and Houston, he is back on the trail, working to define Mr. Gillum — who supports single-payer health care and abolishing the Immigration and Customs Enforcement agency — as too extreme for a state that sits squarely in the middle of the political spectrum.