On CNN, when asked what he would do if Mr. Trump directed him to reopen New York, Mr. Cuomo said, “If he ordered me to reopen in a way that would endanger the public health of the people of my state, I wouldn’t do it.”

[Get the latest news and updates on the coronavirus in the New York region.]

N.Y.C.’s death toll soars past 10,000 as count is revised.

New York City, already the epicenter of the coronavirus outbreak, sharply increased its death toll on Tuesday, by more than 3,700, after officials said they were now including people who had never tested positive for the virus but were presumed to have died of it.

The new figures, released by the city’s Health Department, drove up the number of people killed by the virus in New York City to more than 10,000, and appeared to increase the overall U.S. death count by 17 percent, to more than 26,000.

The numbers brought into clearer focus the staggering toll the virus has already taken on the largest city in the United States, where deserted streets are haunted by the near-constant howl of ambulance sirens. Far more people have died in New York City per capita than in Italy, the country with the most deaths except for the United States.

The revised death toll has also underscored shortcomings in testing that have hamstrung city and state officials since the beginning of the outbreak. The number of tests available has been inadequate, and until now, only deaths where a person had tested positive were counted.

[Coronavirus in New York: A map and the case count.]

Should N.Y. get $12,000 per case if Nebraska gets $379,000?

Mr. Cuomo has repeatedly complained that Congress shortchanged New York in the $2 trillion economic stabilization package it passed last month.

Among the governor’s grievances: States like Nebraska, Minnesota and Montana, he recently said, were getting far more money per virus case than New York.