As Mr. Donovan, the city’s lone Republican congressman, faces a feisty challenge in the general election from the Democratic candidate, Max Rose, Mr. Donovan seems to be moving toward the center. In a district that has more registered Democrats than Republicans, yet skews heavily conservative, the question now is whether Mr. Trump’s support may hurt more than it helps in November.

The issue is one that is playing out across the country, but is particularly germane in New York City, where Mr. Trump is deeply unpopular.

“I don’t think he’ll be talking about that bill to put Trump’s mug in every post office in America,” said Richard Flanagan, a professor of political science at the College of Staten Island. He was referring to legislation that Mr. Donovan had sponsored to ensure that every post office displayed a photograph of the president.

“It’s a scramble to the middle; it really is,” Professor Flanagan said.

Similarly, Mr. Rose, a 31-year-old Army veteran and former health care executive, is trying his best to court the center by winning over unaffiliated voters, as well as Republicans disenchanted with Mr. Trump. In one of two recent television ads, he even goes after a stalwart of his own party, Mayor Bill de Blasio.

“Look,” he says into the camera from a sidewalk in the district. “I’m running against Dan Donovan. But the truth is, he’s not the only one doing a lousy job. Mayor de Blasio acts like Staten Island doesn’t even exist. And we need to get rid of all the leadership in D.C. — Republican and Democrat.”