If you doubt the power of Donald Trump’s endorsement, look no further than the Ohio Senate race.

Since April 15, when Trump backed J.D. Vance in the Republican primary, the venture capitalist and author of “Hillbilly Elegy” has zoomed to the top of the public polls. Vance jumped from 11 percent of likely voters in March to 23 percent now, according to Fox News.

The real test of Trump’s party boss mojo, however, is fast approaching: actual elections, beginning with Ohio’s on Tuesday. Trump has endorsed candidates in at least 40 Republican primaries that are taking place in May, my colleague Alyce McFadden has tabulated. Most of these contests involve an incumbent who faces no serious challenger. But in statewide races from Pennsylvania to Georgia and Idaho to North Carolina, Trump’s imprimatur could prove decisive.

Republicans are watching these races closely for signs that Trump’s hold over the party is waning. Privately, many G.O.P. operatives view the former president as a liability. And while he has shown a unique ability to energize the party’s base and turn out new voters, those operatives are still dreading the likelihood that he runs again in 2024, anchoring candidates up and down the ballot to an erratic, divisive figure who was rejected by swing voters in 2020.

Everyone knows Trump still has juice. But nobody is sure just juice.

“The risk for Trump is that if the candidates he has endorsed end up losing, his influence over Republican primary voters looks substantially diminished,” said Whit Ayres, a longtime Republican pollster.

Already, there are signs of what one G.O.P. strategist called a “re-centering” of Republican politics — with Trump as the party’s strongest voice, but no longer its sole power broker.

In Alabama, he withdrew his endorsement of Representative Mo Brooks, an ardent Trump loyalist who has floundered as a Senate candidate. Senator Ted Cruz of Texas, Trump’s top 2016 foe, has endorsed his own slate of candidates, as have conservative groups like the Club for Growth. And Ron DeSantis, the governor of Florida, is drawing rapturous receptions within the party as he gears up for a likely presidential run in 2024.

The stakes for American democracy are high. In Georgia, Trump is trying to unseat Gov. Brian Kemp and Secretary of State Brad Raffensperger, two Republicans whose refusal to help overturn the 2020 election results have made them the former president’s top targets. In both cases, Trump is backing challengers who have embraced his false narrative of a stolen election.

Trump’s endorsement is no magic wand. In a recent poll by Quinnipiac University, 45 percent of Republicans said they would be more likely to vote for a candidate endorsed by Trump, whereas 44 percent said it would make no difference.

“Every candidate has to lose or win their race themselves,” cautioned Ryan James Girdusky, an adviser to a super PAC supporting Vance.

With that caveat in mind, here’s a look at the key primaries to watch:

Ohio Senate, May 3

Vance was looking wobbly before Trump’s endorsement. His fund-raising and campaign organization were anemic; his past comments, such as his comparison of Trump to “cultural heroin,” were hurting him.

Now, G.O.P. strategists largely expect that Vance will win the primary. Support for Mike Gibbons, a businessman who spent more than $13 million of his own money on ads, is crumbling. Most of his voters appear to be migrating toward Vance rather than Josh Mandel, the other leading candidate in the race, said Jeff Sadosky, a former political adviser to Senator Rob Portman, who is retiring.

But Mandel, a well-known quantity in Ohio conservative politics, appears to be holding his ground.

“If Vance wins, it’ll be because of the Trump endorsement,” said Michael Hartley, a Republican consultant in Columbus who is not backing any of the candidates.

Pennsylvania Senate and governor, May 17

In some ways, Pennsylvania offers the purest test of Trump’s appeal.

Trump recently endorsed Mehmet Oz, the celebrity doctor, in the Republican Senate primary. But unlike in other states, the public polls haven’t moved much. Democratic strategists still see David McCormick, a wealthy former hedge fund executive and the other leading Republican candidate, as a potent threat.

“No one here thinks it’s locked up,” said Christopher Nicholas, a Republican consultant in Harrisburg, though the Oz campaign’s internal polling has shown a shift in the doctor’s favor.

Trump has yet to endorse a candidate for governor here, but his shadow looms large. He issued an anti-endorsement to Bill McSwain, a former U.S. attorney who served in the Trump administration, calling him “a coward, who let our country down” by not stopping “massive” election fraud in 2020.

Two other candidates are ardent backers of his stolen election claims: former Representative Lou Barletta, whose campaign is managed by former Trump advisers; and Doug Mastriano, a state lawmaker and retired colonel who helped organize transportation to the “Stop the Steal” rally on Jan. 6.

Georgia, May 24

The biggest test of Trump’s influence will come in Georgia, where control of the machinery of democracy itself is on the ballot.

It was Georgia where Trump pressured the state’s top elections official to “find 11,780 votes” to overturn the presidential election results, a phone call that is under investigation.

Trump is hoping to oust Raffensperger, the secretary of state, who was on the receiving end of that phone call. The former president has backed Representative Jody Hice, who supports Trump’s debunked election fraud claims. Court documents released by the House committee investigating the Jan. 6 riot place Hice at a meeting at the White House to discuss objections to certifying the election.

In the governor’s race, Trump dragooned former Senator David Perdue into trying to unseat Kemp, the incumbent.

Perdue, who lost to Jon Ossoff, a Democrat, in 2021, has dutifully made 2020 the theme of his campaign.

But polls show Kemp comfortably ahead, suggesting that dwelling on the past is not a path to victory despite the power of Trump’s endorsement.

Lightning round

A few other primaries we’re watching:

• May 10: In West Virginia’s Second Congressional District, redistricting has pitted two Republican incumbents against each other. Trump endorsed Representative Alex Mooney, who voted against the bipartisan infrastructure law, while Gov. Jim Justice is backing Representative David B. McKinley, who voted for it.

• May 17: In North Carolina, Trump’s preferred Senate candidate, Representative Ted Budd, is surging in the polls against former Gov. Pat McCrory and Representative Mark Walker.

• May 17: Gov. Brad Little of Idaho faces a primary challenge from a field that includes his own lieutenant governor, Janice McGeachin, who has Trump’s endorsement. The two have feuded bitterly, to the point where McGeachin issued her own executive orders while Little was traveling out of state. McGeachin also has a history of associating with extremists. In February, she gave a virtual speech at an event sponsored by white nationalists, leading to calls for her resignation.

Alyce McFadden contributed reporting.

What to read

• Reporting from Columbus, Ohio, Jazmine Ulloa notes a new fixation in G.O.P. messaging: the baseless claim that unauthorized immigrants are voting.

• Jonathan Weisman, from Toledo, Ohio, reports that Democrats are in jeopardy because they can no longer rely on firm support from unions.

• Patricia Mazzei explores how, under Gov. Ron DeSantis, Florida has become a laboratory for right-wing policies.

• Emily Cochrane spoke with Senator Lisa Murkowski of Alaska about whether her centrist credentials will appeal to Republican voters in November.

how they run

This Ohio House race has everything

By Leah Askarinam

It’s a shining example of Democrats’ challenges in 2022, the confusion caused by whiplash over new congressional maps and, yes, the power of a Trump endorsement: This is the race for Ohio’s 13th Congressional District.

The district — whose lines are changing and whose current representative, Tim Ryan, is running for Senate — is one of just a few in Ohio expected to be competitive in the fall.

President Biden would have carried this newly drawn district by just three percentage points, making it a must-win for Democrats as they face challenges in maintaining their House majority.

“If 2022 is as bad for Democrats as most everybody else, myself included, expects it to be, Republicans will flip this district,” said Ryan Stubenrauch, a Republican strategist in Ohio who grew up in the area.

An added wrinkle is that the boundaries of the district aren’t technically final: Ohio’s redistricting process has been tied up in the courts, and the State Supreme Court could still rule against the current maps. But most experts believe that the lines will remain in place through the general election.

For Democrats, the primary election on Tuesday should be straightforward. Emilia Sykes, a state representative and former minority leader, will be the only Democrat on the ballot. The Sykes name is well known in the Akron area, where her father, Vernon Sykes, remains in the state legislature. His wife, Barbara, also once served in the state House.

On the Republican side, Trump endorsed Madison Gesiotto Gilbert, a conservative commentator who worked on Trump’s campaigns in 2016 and 2020. She faces Shay Hawkins, a Republican who narrowly lost a state House race in 2020. He’s the only candidate who has aired a campaign TV ad, but he trails Gilbert in fund-raising. A third Republican to watch, Gregory Wheeler, has an endorsement from The Plain Dealer.

No candidates have had much time to make a mark. They learned their district lines — tentatively — just a few weeks before early voting started. And with the primaries split, with state legislative voting postponed to later this year, turnout is a big question.

Some of the usual efforts to inform voters about important dates, like when to register and the deadline for early voting, didn’t happen this year with details in flux, said Jen Miller, executive director of the League of Women Voters of Ohio, which is in litigation over the maps in front of the Supreme Court.

“The delay and the fact that we have to have a second primary for the State House maps is really confusing for voters,” Miller said.

— Blake & Leah