A series of May primaries represent the biggest test yet of Donald Trump’s post-presidential influence on the Republican electorate, with the outcomes carrying implications for his potential third White House campaign and control of Congress.

The fate of several of Mr. Trump’s highest-profile endorsements will be determined during a month that features a dozen primaries from coast to coast. GOP contests for U.S. Senate in Ohio on Tuesday and the battleground states of Georgia, North Carolina, Ohio and Pennsylvania in the following weeks will be among the most closely watched, as well as a Republican primary for governor in Georgia, where Mr. Trump badly wants incumbent Brian Kemp defeated.

In the 15 months since leaving office, the former president has endorsed 150 candidates for offices ranging from state legislatures to the Senate, according to the election data website Ballotpedia.

The large number—a level of involvement in party politics not seen from a former president for at least a century—is part of an effort to remain relevant for a possible second run against President Biden in 2024, something Mr. Trump has strongly hinted he will do. He has often awarded his support to those who display loyalty to him, including agreeing with his false claims of widespread election fraud in the 2020 presidential election.

“Up until now, everything about Trump has been speculation, with a little bit of polling data,” said Scott Reed, a Republican and former chief political strategist for the U.S. Chamber of Commerce. “But by the end of May, we will have hard data on him from these primaries.”

Mr. Reed said he will especially be watching the Senate primaries in Ohio and Pennsylvania. “Those are the two that will be the barometer for telling Trump’s strength with the GOP electorate,” he said.

Mr. Trump’s deep involvement in GOP primaries has left some establishment Republicans like Mr. Reed worried that some of his selections—if they are successful in their primaries—could prove to be flawed in the general election, when candidates must win over independent voters and the former president’s seal of approval could prove to be harmful.

Polling suggests an endorsement from Mr. Trump can be somewhat helpful among GOP voters. A Quinnipiac University survey released Wednesday showed 45% of Republicans say they are more likely to vote for a Trump-endorsed candidate, while 8% say they are less likely and 44% say it doesn’t make a difference.

Supporters for J.D. Vance, who is running against six other candidates in the Senate primary in Ohio and who lobbied for Mr. Trump’s support.

Photo: GAELEN MORSE/REUTERS

May 3: Indiana and Ohio

The first verdict on Mr. Trump’s ability to serve as a party king maker will come in Ohio, where he has endorsed venture capitalist and author J.D. Vance in one of the party’s most contentious primaries.

Mr. Vance in 2016 called Mr. Trump “unfit for our nation’s highest office.” But he subsequently cozied up to the former president and lobbied hard for his backing, as did virtually all others in the race.

Limited public polling suggests a close contest in a race with seven candidates on the ballot, with four of them in double digits in a RealClearPolitics.com polling average: Mr. Vance, former State Treasurer Josh Mandel, investment banker Mike Gibbons and Matt Dolan, a state senator and former prosecutor.

Among the top candidates, Mr. Dolan is alone in not chasing Mr. Trump’s endorsement, and he has campaigned as a more traditional Republican, giving an option for voters who might not like how closely the others have tied themselves to the former president.

The Ohio primary, for a seat now held by retiring Republican Sen. Rob Portman, has seen the most ad spending on the GOP side so far this election cycle. More than $61 million had been spent by the candidates and their allies through April 28, data from ad-tracking firm AdImpact shows.

“It’s the most I’ve ever seen on TV for a statewide race here,” said Mark R. Weaver, a GOP strategist who lives in Ohio.

Mr. Weaver said Mr. Trump’s endorsement has boosted Mr. Vance, but lost some of its force because it came after early voting had already started. “Many had already voted before his announcement, or picked a candidate before Trump announced,” he said.

May 10: Nebraska and West Virginia

In Nebraska’s race for governor, the former president is backing Charles Herbster, a farmer and chief executive of a manufacturing company who has been accused of inappropriately touching numerous women, including a Republican state senator. Mr. Herbster, who had been advised by former Trump campaign manager Corey Lewandowski, has denied the allegations and was scheduled to appear with the former president in the state on Sunday evening.

Charles Herbster, Republican candidate for governor of Nebraska, is scheduled to appear with Mr. Trump on Sunday night.

Photo: Tristan Wheelock/Bloomberg News

In West Virginia, two House Republican incumbents are competing because the state lost a seat in reapportionment. Mr. Trump backs Rep. Alex Mooney over Rep. David McKinley, who voted for bipartisan infrastructure legislation the former president opposed and for a commission to investigate the attack on the Capitol on Jan. 6, 2021, by Mr. Trump’s supporters.

May 17: Idaho, Kentucky, North Carolina, Oregon and Pennsylvania

The highest-profile contest on May 17 is the GOP primary for Senate in Pennsylvania. Mr. Trump endorsed celebrity surgeon Mehmet Oz, widely known as Dr. Oz, after the first candidate he backed dropped out amid allegations of domestic abuse. David McCormick, a former chief executive of hedge-fund giant Bridgewater Associates, is the other top candidate in the race.

Television personality Mehmet Oz has Mr. Trump’s endorsement in Pennsylvania’s high-profile GOP primary for Senate.

Photo: HANNAH BEIER/REUTERS

Pennsylvania and West Virginia reveal cracks in Mr. Trump’s grip on the party, even among people who once worked for him. Several of Mr. Trump’s former White House aides are working for Mr. McCormick. Former Secretary of State Mike Pompeo, a potential 2024 presidential candidate, has endorsed Mr. McCormick in Pennsylvania and Mr. McKinley in West Virginia.

North Carolina’s Senate primary, where Mr. Trump decided to back Rep. Ted Budd over former Gov. Pat McCrory, will be another closely watched May 17 contest. Mr. Budd leads in the RealClearPolitics.com polling average and has the backing of Club for Growth Action, a super PAC that is the race’s largest advertising spender so far.

Mr. Trump is also backing Rep. Madison Cawthorn in a crowded House race in North Carolina. Mr. Cawthorn, known for his brashness and for being the youngest member of Congress at 26 years old, recently saw undated photos of him wearing women’s lingerie published by Politico. He has said they were from a game on a cruise before he ran for Congress.

In Idaho’s May 17 primary, Mr. Trump has endorsed Republican Lt. Gov. Janice McGeachin over incumbent Republican Gov. Brad Little. Both candidates sought his backing during visits to his Mar-a-Lago club in Florida.

May 24: Alabama, Arkansas and Georgia

Georgia’s gubernatorial primary includes one of Mr. Trump’s riskier bets. Mr. Kemp, the incumbent, has a large polling lead over David Perdue, a defeated U.S. senator recruited by Mr. Trump who has echoed the former president’s false claims of widespread election fraud in the 2020 election.

Georgia Gov. Brian Kemp has a large polling lead over a challenger recruited by Mr. Trump to run against him in the GOP gubernatorial primary.

Photo: Branden Camp/Associated Press

Mr. Trump has been fixated on trying to defeat Mr. Kemp because of his unwillingness to help throw out 2020 election results that showed Mr. Trump had lost Georgia. The governor didn’t have the authority to take such action and has generally remained quiet on the topic.

On the campaign trail, Mr. Kemp has avoided mentioning Mr. Trump. At a recent event in northern Georgia, the Republican chairman of the Lumpkin County Board of Commissioners took the same approach, while also appearing to comment on Mr. Trump.

“I’m an independent thinker and I don’t need a politician that’s been defeated telling me who I need to vote for for the next governor of the state of Georgia,” Chris Dockery, who is backing Mr. Kemp, said to applause.

Later, in an interview, Mr. Dockery said “most rational people” don’t blame Mr. Kemp for Mr. Trump’s 2020 defeat in Georgia.

“His influence is dying,” Mr. Dockery said. “People are starting to focus on the issues and not so much on the politics of Donald Trump.”

Trump spokesman Taylor Budowich predicted the demise of so-called Republicans in name only. “In American politics, there is nothing more consequential than the endorsement of President Donald J. Trump and the only people disputing that reality are irrelevant RINOs breathing their final breaths before their political extinction,” he said in a statement.

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Herschel Walker, a former professional football player who has acknowledged a battle with mental illness that included playing Russian roulette with a loaded gun, is likely to deliver a win for Mr. Trump in Georgia’s GOP senate primary. Surveys, however, suggest a possible tight general election race with the well-financed Democratic incumbent, Sen. Raphael Warnock.

Mr. Trump has no endorsement in place for Alabama’s Senate primary following his decision to rescind his backing of Mo Brooks. The move came after the congressman appeared to be struggling in the race and had moved on from emphasizing the former president’s claims of a stolen 2020 election.

While May will offer a robust look at Mr. Trump’s influence with the GOP base, his full win-loss record won’t be known until more primaries are completed this summer, including high-profile contests Aug. 16 in Wyoming and Alaska, where he is supporting challengers to incumbent Republicans.

—Cameron McWhirter contributed to this article.

Write to John McCormick at mccormick.john@wsj.com