“The president obviously has very strong feelings about his form of politics,” said Gov. Bill Haslam of Tennessee, who is the group’s chairman, acknowledging he had not known Mr. Trump planned to back Mr. Kemp. “You’ve seen them engage in primaries where traditionally the White House has not engaged. Our focus at the R.G.A. has always been on making certain we can win the general election.”

Privately, the governors and their allies, some of whom are gathered here in the New Mexico capital for the annual summer meeting of the National Governors Association, expressed even more irritation about the president’s endorsement in Georgia.

They are frustrated on several fronts: because Mr. Trump’s out-of-the-blue tweet in support of Mr. Kemp was only the latest example of him intervening in races with no warning to the party infrastructure; because he continues to side with candidates they fear could prove weaker in the general election; and because Mr. Trump had assured Republican governors in private this year, at the last meeting of the National Governors Association, that he would refrain from involving himself in contested primaries, according to an official present for the conversation.

Mr. Trump’s role in the Georgia race marks the second time in two months where he has offered a full-throated endorsement of a candidate who may complicate his party’s chances to hold onto a governorship in November. The winner of Tuesday’s runoff will face Stacey Abrams, who is vying to become the first black woman to serve as governor. Private polling conducted for both parties has found Ms. Abrams, the Democratic nominee, in a position to win the general election, and strategists on both sides of the race see Mr. Kemp as a riskier choice for Republicans.