New York City Mayor Bill de Blasio on Wednesday took a subtle shot at Gov. Andrew Cuomo over the state's sluggish COVID-19 vaccine rollout, pushing his fellow Democrat to broaden the eligibility criteria to include older New Yorkers and all essential workers. 

"We need the freedom to vaccinate," de Blasio said during a news conference. "This city needs the freedom to vaccinate the highest number of people possible. We need the freedom to vaccinate because with that freedom and flexibility, we can reach so many people."

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The effort to vaccinate millions of New Yorkers is off to a slow start, frustrating city and health officials as the virus surges nationwide and a variant of a new strain that's more contagious has been detected in the city. 

New York will set up 250 city-run vaccination sites this month in an effort to speed up the process and administer 1 million doses by Jan. 31, de Blasio said earlier this week.

Only health care workers and nursing home residents and staff members are currently being vaccinated in New York. De Blasio has said it’s time to widen eligibility to include people older than 75 and essential workers such as grocery store workers, firefighters, and teachers.

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"My central concern right now is folks over 75," he said. "Look, folks over 75 are the most vulnerable in this crisis. Right now, we are not allowed to vaccinate them according to state rules. We need the freedom and flexibility so that we can start vaccinating folks over 75 right away."

Only about 100,000 people received vaccines in New York City during the first three weeks after the first vaccine was approved for emergency use, but de Blasio has pledged that the city will administer 400,000 vaccines a week by the end of this month.

"Let's get to all essential workers -- whether you work in a grocery store, food service or you're a police officer or a firefighter or an educator, we need to reach all those essential workers as quickly as possible," he said. "We are looking for that freedom."

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Cuomo, meanwhile, has threatened to fine hospitals that don't administer their vaccines quickly enough. 

"If you don’t want to be fined, just don’t participate in the program," he said Monday. "It’s not a mandatory program."

The Associated Press contributed to this report