President Trump has spent the past year using the federal government to target those he considers his political enemies. Now, in a series of primaries this month, he is turning his ire on fellow Republicans by asking voters to punish those who have crossed him.

Contests in Indiana, Louisiana and Kentucky in the coming weeks all feature Trump-backed challengers trying to oust Republican incumbents who have earned Mr. Trump’s ire. The fights have soaked up energy and cash from the president’s allies and his political operation, pulling attention away from the broader battle with Democrats over control of Congress this fall.

In Indiana, Mr. Trump hopes to oust a half-dozen Republican state senators who defied him by defeating a redistricting effort that would have drawn the state’s two Democratic House members out of power. In Louisiana, he is backing a challenger to a Republican senator, Bill Cassidy, who voted to convict him in his second impeachment trial after the Jan. 6, 2021, Capitol riot. In Georgia, Mr. Trump is supporting a candidate for governor who is running against the Republican secretary of state, Brad Raffensperger, who refused to help the president “find” votes for him in the aftermath of the 2020 election.

And in Kentucky, Mr. Trump is invested in unseating one particularly irksome Republican congressman who has refused to vote for the president’s full agenda.

“We’ve got to get rid of this loser,” Mr. Trump said at a March rally in Representative Thomas Massie’s northern Kentucky district, a deeply conservative swath encompassing suburbs of Cincinnati.

Mr. Massie has openly challenged Mr. Trump in a way that no other congressional Republican seeking re-election this year has done, even as he has stressed that he sided with Mr. Trump on roughly 90 percent of votes.