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.