On Monday, New York Governor Andrew M. Cuomo is under intensifying pressure to overhaul the process of coronavirus vaccinations as the onetime center of the pandemic is facing a growing crisis over its slow pace, with deaths continuing to rise in the second wave.

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In New York City, a city with more than eight million populaces, only roughly 110,000 people have received the first of two coronavirus vaccine doses necessary to help prevent severe disease cases. The small number of vaccine recipients is most striking as it is only a quarter of the total number received by the city.

New York concerns over coronavirus vaccinations that echo glitches conveyed across the nation's slow rollout, and here comes a 60-year-old man who is the first confirmed case of a more transmissible coronavirus variant. The man was recovering in the northern city of Saratoga Springs, but Mr. Cuomo said evidence of community spread was earlier indicated. Mr. Cuomo said, "I think it is much more widespread than people know."

The planned 19.5 million resident's coronavirus inoculation got complicated after confirming the variant in New York, with criticism opening to mount over the rollout. On Monday, the Cuomo administration was called by Mayor Bill de Blasio to allow the broader array inoculation of essential workers and 75 and older New Yorkers.

On Monday, at a news conference, Mr. de Blasio said, "There's lots more we can do if we have both those categories approved." The state has to distribute coronavirus vaccine deliberately until Monday, and the coronavirus vaccinations were nearly exclusively given to group home residents, health care workers, and those from nursing homes.

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In the state's initial guidance, careful tactics were also apparent to determine who among healthcare employees should be prioritized for coronavirus vaccines. The state had directed clinics and other medical facilities to rank personnel through a matrix of age, comorbidities, profession, and the facility section where the person is assigned.

That matrix guidance was no longer current, said State officials, on Monday as the governor approved a broader healthcare provider swath, and others possibly exposed to the coronavirus to receive the shot, including primary care doctors and pediatricians.

Any notion that his administration was accountable for not dispensing more coronavirus vaccines was rejected by Mr. Cuomo, emphasizing that the problem was a local issue.  He urges leaders who oversee public hospital systems to take "personal responsibility" for their performance. In Albany, the governor said, "They have to move the coronavirus vaccine, and they have to move the vaccine faster."

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The governor also threatened hospitals that if they did not hastily increase vaccination pace, he will fine hospitals up to $100,000 and readdress future vaccines to other hospitals.

Mr. Cuomo said, adding, "We want those vaccines in people's arms. This is a very serious public health issue."

Mr. Cuomo also said with worries about fraud circulating and inoculating anxious residents, and he would recommend legislation to impose criminal charges for facilities or health care providers that did not follow procedures on who is vaccine eligible, which he described as "like gold to some people."