Public officials touted the success of the mandate in raising vaccination rates. When it was first announced by then-Gov. Andrew M. Cuomo on Aug. 16, only 70 percent of nursing home staff had received at least one vaccine dose and 77 percent of hospital staff were fully vaccinated.

In a shift, the governor’s office on Tuesday estimated the total number of hospital workers affected by the mandate at 519,000 statewide, up from its estimate of 450,000 when the mandate was announced.

“People who are on the fence benefit from these mandates, bluntly, as a way to make this decision,” said Mayor Bill de Blasio, who has also required that everyone working in New York City public schools have at least one vaccine dose by next Monday.

While the rates were high among health care workers as a whole, however, low vaccination totals at some facilities sent a warning signal. As of Sunday, fewer than 65 percent of staff members had received at least one vaccine dose at several nursing homes, including The Plaza Rehab and Nursing Center in the Bronx, and Hopkins Center for Rehabilitation and Healthcare in Brooklyn, state data showed.

What to Know About Covid-19 Booster Shots

The F.D.A. authorized booster shots for a select group of people who received their second doses of the Pfizer-BioNTech vaccine at least six months before. That group includes: vaccine recipients who are 65 or older or who live in long-term care facilities; adults who are at high risk of severe Covid-19 because of an underlying medical condition; health care workers and others whose jobs put them at risk. People with weakened immune systems are eligible for a third dose of either Pfizer or Moderna four weeks after the second shot.

The C.D.C. has said the conditions that qualify a person for a booster shot include: hypertension and heart disease; diabetes or obesity; cancer or blood disorders; weakened immune system; chronic lung, kidney or liver disease; dementia and certain disabilities. Pregnant women and current and former smokers are also eligible.

The F.D.A. authorized boosters for workers whose jobs put them at high risk of exposure to potentially infectious people. The C.D.C. says that group includes: emergency medical workers; education workers; food and agriculture workers; manufacturing workers; corrections workers; U.S. Postal Service workers; public transit workers; grocery store workers.

For now, it is not recommended. Pfizer vaccine recipients are advised to get a Pfizer booster shot, and Moderna and Johnson & Johnson recipients should wait until booster doses from those manufacturers are approved. ​​The F.D.A. is planning to allow Americans to receive a different vaccine as a booster from the one they initially received. The “mix and match” approach could be approved once boosters for Moderna and Johnson & Johnson recipients are authorized.

Yes. The C.D.C. says the Covid vaccine may be administered without regard to the timing of other vaccines, and many pharmacy sites are allowing people to schedule a flu shot at the same time as a booster dose.

At the Erie County Medical Center in Buffalo, 20 percent of staff at its affiliated nursing home, Terrace View, were placed on unpaid leave on Monday for refusing to get vaccinated, a spokesman said. The hospital said it was doing its best to make up for the reduction by transferring staff in from other facilities, reducing beds at the nursing home and suspending some elective surgeries at the hospital.

The medical center had been predicting 400 staff departures, but in the end, only 276 unvaccinated workers were placed on leave. Still, the facility remained in crisis mode, because in the weeks before the mandate came into effect, the hospital already had 400 job vacancies and a record number of patients.

The hospital was having trouble, said Tom Quatroche, its president, because it could not discharge patients to nursing homes and rehab centers that were experiencing their own staff shortages.