The latest cases illustrate how the authorities still know little about how long the virus has been present here, or how many people have already been infected.

“It’s hard to conclude how long it has been circulating in our environment,” Dr. Michael Phillips, the chief hospital epidemiologist for the N.Y.U. Langone Health System, said in an interview Wednesday, before the two most recent cases in the city were announced, but after the detection of the Westchester cases.

Dr. Phillips said that as local capacity to test for the virus continues to expand in the coming days, much more would be learned about the prevalence of the virus here. “There’s no doubt that the ability to test readily may rapidly change our appreciation for the extent of the disease in a place like New York,” he said.

The Long Island patient is being treated at N.Y.U. Winthrop Hospital in Mineola. Hospital officials said on Thursday that the diagnosis was possible because the C.D.C. had recently expanded the criteria for who was eligible for testing.

Over the last few weeks, various state and city officials have urged the C.D.C. to broaden testing parameters and allow local testing to expedite the ability to detect cases. Last weekend, the Wadsworth Lab in Albany began administering coronavirus tests after receiving permission from the federal government; New York City began its own tests earlier this week.

Mr. de Blasio on Thursday pressed the C.D.C. to increase the city’s “supply of Covid-19 test kits and expedite the approval of any testing approaches developed by private companies.”

“The federal government was very late to the dance in getting tests out to localities,” the mayor said. “They’re still late in terms of getting the volume we need.”