“It would be nice if he passed a piece of legislation that actually helped the State of New York,” Mr. Cuomo said on Sunday.

The governor’s soft touch with the president would seem to be a departure from his style in Albany, where he has been in office for nearly a decade, and is known as the consummate political animal who thrives on conflict.

But Mr. Cuomo is also a proudly pragmatic deal-maker who has readily and regularly worked with Republicans when that party controlled the State Senate, infuriating some fellow Democrats, especially progressives. And in dealing with the president, the governor has seemed to refine that strategy, recognizing that Mr. Trump’s political antennae are fine-tuned to detect both praise and put-downs.

“The trick to Trump is not to embarrass him in public,” said Hank Sheinkopf, a veteran Democratic political consultant who has known both men for decades. “Being a Trump whisperer means he can criticize him obliquely, by talking about the national government’s failure to respond.

“But he can’t attack him personally,” Mr. Sheinkopf continued, “because then he loses any opportunity to talk to him behind the scenes to get anything done.”

Mr. Cuomo’s tactics have seemingly paid off in tangible ways. In the month since the virus was first discovered in New York, the governor has finagled or finessed the creation of thousands of federally financed hospital beds; the federal authorization of labs in the state to run coronavirus tests; and the arrival on Monday of a federal hospital ship, U.S.N.S. Comfort, which the state hopes will help relieve pressure on local hospitals, though it has been largely underused thus far.