Donald J. Trump barreled into Georgia vowing to marshal voters against his enemies and punish Republicans who crossed him in 2020. Instead, Georgia voters punished him for meddling in their state.Mr. Trump picked losers up and down the ballot, most strikingly missing the mark on a third governor’s race in three weeks. The dismal record, particularly for chief executives, illustrates the shortcomings of Mr. Trump’s revenge tour.Since leaving the White House, and the structure it provided, the former president has erratically deployed his political power, often making choices on a whim or with little clear path to execution. That approach has repeatedly left him empty-handed and raised new doubts about the viselike grip he has held on the Republican Party.In Georgia, Mr. Trump tried to wipe out a triumvirate of Republican statewide officeholders who refused to help overturn the 2020 presidential results: Gov. Brian Kemp, Secretary of State Brad Raffensperger and Attorney General Chris Carr. But the three men all coasted to victory — and handed Mr. Trump a stinging rebuke in a state that has become one of the nation’s most important presidential battlegrounds.Henry Barbour, a Republican National Committee member from Mississippi, said Mr. Trump’s endorsements this year had been “driven by who he dislikes and whoever’s running against them.”“Sometimes that may work out, but I think as we see in Georgia, it’s very unlikely to,” Mr. Barbour said.Mr. Trump’s poor showing in state capitals — his endorsed candidates for governor have now lost as many races as they’ve won this year — can be partly blamed on the degree of difficulty of his undertaking. Successful campaigns for governor often must be precisely tailored to address nuanced regional and local issues. House and Senate bids — where Mr. Trump’s endorsement record as yet is nearly unblemished — can more easily harness national political winds.Unseating incumbent governors in a primary, as Mr. Trump tried to do in Georgia and Idaho, is even more challenging. According to the Eagleton Center on the American Governor at Rutgers University, governors defeat primary challengers about 95 percent of the time. Two incumbent governors haven’t lost primaries in the same year since 1994.But Mr. Trump has shown the unlikely to be practically impossible when decisions about endorsements for high-profile public offices are based on falsehoods, vengeance and personal pride. His refusal to take a more cautious approach and protect his political capital ahead of a likely 2024 presidential campaign has resulted in unforced errors that could unspool for months.Gov. Brian Kemp in Atlanta on Tuesday night after winning the Republican primary.Nicole Craine for The New York TimesIn Georgia, for example, Republicans have worried about the unnecessary political damage Mr. Trump has inflicted on Mr. Kemp, who will face a rematch in November with Stacey Abrams, the Democratic nominee who lost their 2018 contest by 54,700 votes, or less than one-half of one percentage point. Political control of the governor’s office carries significant influence over election laws and regulations heading into the 2024 voting.“I don’t believe Kemp can do it,” Mr. Trump said during a tele-rally on Monday about the governor’s chances of defeating Ms. Abrams. “He’s got too many people in the Republican Party that will refuse to vote. They’re just not going to go out.”Mr. Trump’s loss in Georgia also meant a major victory for the Republican Governors Association, which has circled the wagons around incumbents and resisted the former president’s attacks on their members.The group spent $5 million on Mr. Kemp’s race and dispatched a cavalry of current and former governors to campaign for him, including two potential challengers to Mr. Trump in 2024: Chris Christie of New Jersey and former Vice President Mike Pence, who, like Mr. Kemp, refused to help Mr. Trump overturn the 2020 election.Whether the Georgia results will provide a toehold for a challenge to Mr. Trump’s supremacy in the party remained unclear, but signs that he has lost some political altitude have been unmistakable throughout the 2022 primary season.Mindful that potential 2024 presidential rivals are watching for openings against him, Mr. Trump has been toying for months with announcing his candidacy ahead of the midterm elections this year, according to people who have spoken with him.Earlier talk of similar moves went nowhere, including a “Draft Trump” movement floated by Senator Lindsey Graham of South Carolina shortly after Mr. Trump left office and an idea to announce an exploratory committee as recently as March.But Mr. Trump has spoken to aides recently about declaring his candidacy this summer as a way to box out other candidates. Other advisers said he viewed an announcement as a way to link himself to the success that Republicans expect in the midterm elections this November.Herschel Walker with former President Donald J. Trump at a March rally in Commerce, Ga.Audra Melton for The New York TimesMr. Trump had a smattering of success on Tuesday night, notably with his former White House press secretary Sarah Huckabee Sanders, who is almost certain to become the next governor of Arkansas after winning the Republican primary. And Herschel Walker, the former football star whom Mr. Trump urged to run for Senate, easily won his Georgia primary. Still, they were exceptions, and faced weak opposition.In a statement, Taylor Budowich, a spokesman for Mr. Trump, ignored the string of defeats Tuesday night, calling it “another huge night of victories for his endorsed candidates.”In an interview last week, Mr. Trump defended his endorsement record, saying he had backed candidates he believed in, not just those he expected to win. He pointed to J.D. Vance’s win in the Ohio Senate primary, Mr. Trump’s biggest victory of the primary season so far.There, Mr. Trump acted after polling late in the race suggested his endorsement could make the difference for Mr. Vance, who was behind in the polls at the time — and his announcement propelled Mr. Vance to a decisive victory.But that deliberate decision-making was a departure from the more scattershot approach seen in Mr. Trump’s endorsements for governor.In Nebraska, where Mr. Trump’s endorsed candidate, the agriculture executive Charles Herbster, was defeated on May 10, Mr. Trump has privately faulted the Fox News host Jeanine Pirro, complaining that she pushed him to back Mr. Herbster, and going so far as to suggest to some people that Ms. Pirro and Mr. Herbster had dated.Understand the 2022 Midterm ElectionsCard 1 of 6Why are these midterms so important? More