The Biden administration’s decision to grant special status to Venezuelan migrants will have no greater impact than in New York City, where thousands of immigrants will soon be eligible to begin legally applying for work and eventually move out of taxpayer-funded shelters.

Mayor Eric Adams and Gov. Kathy Hochul, both Democrats, cast the move as a badly needed reprieve, saying on Thursday that it could help some of the 60,000 migrants in city care move to independent housing more quickly. The policy could also help ensure that other Venezuelans never need the shelters.

Yet even as they celebrated a policy shift after months of intensely lobbying the White House, top New York officials, immigration experts and shelter operators labored to figure out the practical and political impact of a decision many thought President Biden would never make.

Some conceded that it would do little to defray the larger crisis confronting the nation’s largest city — and could eventually worsen it by encouraging more people to immigrate.

“There was a clear call coming from New York City on work authorization, and I want to thank the White House on hearing us,” Mr. Adams said on Thursday, before adding a more downbeat note. He said 10,000 migrants were arriving in the city every month, “and a substantial number are still in our care today.”

In its announcement late Wednesday, the Homeland Security Department said it planned to grant special status to 472,000 Venezuelans who arrived in the country before July 31, protecting them from removal for 18 months and waiving a monthslong waiting period for them to seek employment authorization.

The administration had already extended special protections to close to 250,000 Venezuelans who had arrived in the country before March 2021, as well as to thousands of others from Ukraine, Haiti, Honduras, Nicaragua and other countries. On Thursday, Homeland Security announced that it would extend protections to Afghan immigrants.

The program does not provide a permanent path to legal residence, but has been used by presidents in both parties to grant humanitarian protections to migrants fleeing wars, natural disasters and other violence. The Biden administration said that Venezuelans, who constitute one of the largest sources of migration, fit that description. Venezuelans are also the largest group of the more than 110,000 migrants who have arrived in New York City since spring 2022. But estimates of just how many of those migrants were still in the city fluctuated on Thursday.

Asher Ross, a senior strategist for New York Immigration Coalition Action, said that between 40,000 and 50,000 Venezuelans of working age in the city may qualify for the protections. Mayor Adams and a senior New York homeland security official each said that the number living in shelters was probably smaller, between 10,000 and 20,000.

“If it has bite, that bite will take time to take effect,” said Muzaffar Chishti, a senior fellow with the nonpartisan Migration Policy Institute, predicting the city could even see a short-term bump in migrants seeking shelter. “So New York City still has to be prepared to deal with the housing crisis.”

Elsewhere, there were fierce pockets of opposition to the policy. New York Republicans argued that it would only encourage more Venezuelans and migrants of other nationalities to attempt the arduous journey to cross the border, and warned that the crisis in the country would only worsen until Mr. Biden and Democrats in Congress agree to tougher border policies.

“The president is unwilling either by default or design to confront the crisis at the border — and, this will only make matters worse,” said Representative Marc Molinaro, a moderate Republican from the Hudson Valley. Democrats in power here and in Washington are increasingly nervous that Republicans like Mr. Molinaro could tap into a vein of unease in next year’s elections among voters who might be open to immigration but believe Democrats are bungling the response to the present crush of new arrivals.

News of the new policy spread quickly among Venezuelan migrants on social media. In interviews, many described feelings of relief after months — or longer — living in the shadows, but said they had little information about how to legally claim their new status.

“We’ve been living in fear, and it takes it away,” said Julio Cesar Gonzales, who crossed into the United States with his family in July after journeying for weeks from Caracas. “The fear of not having work, of not eating, of our kids not going to school.”

Mr. Gonzales, 49, said he learned of the new policy on Instagram Thursday morning. He was living in a shelter in Queens with his wife and three young children. With the economy in Venezuela in tatters, he said his family had often gone without food because he could not make ends meet as a taxi driver. “We are now here legal; we can work and get our kids in school, God willing,” he said.

Government officials who will be responsible for administering the policy said it would not be that simple. Migrants will still have to formally apply for temporary protected status. But instead of waiting half a year to file a separate application for work authorization like other asylum seekers, eligible Venezuelans will be able to do so immediately.

The state also plans to begin offering immediate career counseling to those who are eligible and has secured pledges from businesses who are willing to hire workers and delay their start dates until the permits are approved. But others warned of bureaucratic backups that could slow the process, not to mention the challenges even the most successful migrants might face finding housing in a city suffering through an affordability crunch.

“This isn’t a magic wand from Washington that gets bestowed upon you, if you show up from Venezuela,” said Christine Quinn, the president and chief executive of Win, the largest provider of shelters to homeless families with children in the city.

Still, she called it a “game changer” for families ground down by life in shelters.

On Thursday, the mayor and the governor said they would keep pressuring Mr. Biden to help secure funds from Congress to reimburse the city for migrant care and find new ways to accelerate the work authorization process for those not eligible for temporary protected status.

Mr. Adams, in particular, has been warning for weeks that the issue could “destroy” New York City without drastic federal intervention at the border and help to move migrants into legal jobs more quickly. His administration has estimated that its response efforts could cost the city $12 billion over three years, and has already used those figures to justify budget cuts.

But Ms. Hochul, who has opened the state’s arms to migrants, has begun to shift toward a more nuanced tone, discouraging migrants from choosing New York as a destination.

“I will not look a gift horse in the mouth,” Ms. Hochul said on Spectrum NY1. “This is an important first step, and to others that we have to let people know, if you’re thinking of coming to New York, we are truly out of space.”

She also indicated greater openness to taking more drastic measures at home. Right now, New York City is providing free shelter to every migrant who asks for it as part of a decades-old “right-to-shelter” legal mandate.

In an interview late Wednesday on CNN, Ms. Hochul said she agreed with Mr. Adams that it ought to be suspended or altered, a step that would meet fierce resistance from advocacy groups for the homeless and immigrants.

“Never was it envisioned that this would be an unlimited universal right or obligation on the city to have to house literally the entire world,” she said. “We have to let the word out that when you come to New York, we’re not going to have more hotel rooms.”