An influx of migrants arriving in New York City over the last year has stretched city resources and strained political relationships among Democratic leaders grappling with an emerging humanitarian crisis.

But until last week, Gov. Kathy Hochul of New York had stayed largely above the fray.

Now, as New York City’s shelter system barrels past a breaking point, Ms. Hochul is confronting an explosive political test that could define her first full term and have ripple effects for national Democrats.

Critics from the left and right have called on Ms. Hochul to take a more hands-on approach to a crisis that defies easy solutions, saying that the migrant state of emergency — which she announced last spring — needs the same sense of urgency as the Covid pandemic. She has not held frequent briefings on the issue, as she did in past years with the pandemic, and has avoided imposing far-reaching statewide policy, mostly leaving it to Mayor Eric Adams to play the public role of crisis manager.

Her circumspect and rather muted approach to governing is a conspicuous departure from the domineering presence of her predecessor, Andrew M. Cuomo, and it remains to be seen how her less in-your-face style will affect how New Yorkers view this crisis. Ms. Hochul has argued that she has been an active partner in dealing with the crisis all along, even if much of the state’s help has unfolded behind the scenes.

“This is nothing anyone could have anticipated and it’s been an enormous challenge,” she said in an interview on NY1 on Wednesday, adding, “We’re committed to continuing to work with the city to solve what is an enormous crisis. It is a federal crisis as well and we need more help.”

But with migrants continuing to arrive, Mr. Adams recently escalated his calls for the state to get even more involved. The mayor, as well as a coalition of immigration and community groups, has called for Ms. Hochul to implement a statewide relocation program so that upstate counties share the burden of housing migrants. And Mr. Adams and advocates have pressed for executive orders to prevent reluctant municipalities from barring migrants after Republicans in the city’s northern suburbs sued when the mayor tried to bus migrants there in May.

The growing pressure has forced the governor to balance upstate and downstate constituencies with opposing degrees of willingness to accept migrants in their communities.

Moving more migrants outside New York City is bound to spark divisive flare-ups in the suburbs and more conservative parts of the state, fueling attacks from national Republicans who have already harnessed the situation as an election issue talking point and to boost fund-raising.

But failing to relieve the burden on the city, where more than 100,000 migrants have arrived since last year, could further strain the city’s shelter system and housing shortage — and potentially alienate city residents as the crisis deepens.

That dynamic has left many political observers to question how long Ms. Hochul will be able to sustain her hands-off approach and whether she may eventually be forced into taking more aggressive action.

“The reality is that when you’re governor, it’s going to be on you, whether you choose to engage or not,” Doug Forand, a Democratic political consultant, said. “It’s not surprising she doesn’t want to wander into this particular minefield, because it’s really ugly, but in the absence of some form of action, she’s going to have to take those stronger positions soon.”

Most New York Democrats, including Ms. Hochul and Mr. Adams, have largely blamed the federal government, which oversees the nation’s immigration policy, for the spiraling crisis. But that frustration with the Biden administration’s response has intensified calls for Ms. Hochul, a moderate Democrat from Buffalo, to fill the void and take ownership of the crisis.

To be sure, the governor has steered a vast amount of state resources to help the city. She committed over $1 billion in state money, deployed nearly 2,000 National Guard members and helped turn some state-run facilities into shelters. And she has continued to lobby the White House to expedite permits to allow migrants to work legally while their asylum cases are pending.

Her critics, however, say she should do more.

“She could take charge of the situation and limit the ability of local governments to create their own policy around migrant issues,” said Joshua Goldfein, a staff attorney at the Legal Aid Society. “She’s the chief executive of the state, so she should assert her authority to impose a statewide policy.”

“That’s what Andrew Cuomo would have done,” he added, referring to the former governor’s often eager approach to crisis management. (Mr. Cuomo’s former top aide, Melissa DeRosa, criticized Ms. Hochul this week for a “lack of leadership and sheer incompetence.”)

Ms. Hochul rebuffed those calls this week, leading to an exchange of highly critical letters in which the state detailed what it said were shortcomings in the city’s handling of the crisis. Ms. Hochul argued on Wednesday that the city was better equipped than rural counties to house migrants, because of the availability of jobs, public transportation and English-as-a-second-language programs.

“Putting someone in a hotel on a dark, lonely road in upstate New York and telling them they’re supposed to survive is not compassion,” she said in an interview with NY1.

In a statement, Avi Small, a spokesman for the governor, stressed that “it remains appropriate” for the city to care for migrants because newcomers were flocking there as a result of its unique legal requirement to house anyone in need of a bed. The way out of the crisis, he insisted, was getting migrants legal jobs, “not temporarily shuttling individuals outside of New York City against their will.”

The governor’s moves could have political ramifications for Democrats vying to wrest back control of the House from Republicans in 2024 after New York almost single-handedly cost the party its majority last year.

Democrats have begun to fret openly that a failure to resolve the migrant crisis could hurt their chances of reclaiming four House seats on Long Island and in the Hudson Valley that they lost to Republicans.

Howard Wolfson, a former deputy mayor and political adviser to Mayor Michael Bloomberg, described the issue as a “ticking time bomb” for Democrats, adding that busing migrants upstate could backfire and amplify Republican attacks in competitive races.

“There is no question in my mind that the politics of this is a disaster to Democrats,” he said. “This issue alone has the potential to cost Democrats the House, because it is such a huge issue in New York City and the coverage of it is clearly heard and seen by voters in all of these swing districts in the suburbs.”

Indeed, Republicans have already used the barrage of headlines about the issue as fodder to fuel fund-raising efforts and accuse Democrats of prioritizing immigrants over U.S. citizens.

When news broke in June that migrants would be housed at an unused hangar at Kennedy Airport, Representative Anthony D’Esposito, a Republican from Long Island, sent an email to supporters saying that Democrats had “chosen to put politics ahead of the safety of New Yorkers, the aviation industry and its passengers.”

Mr. Adams’s surprise attempt in May to bus migrants to a hotel in Rockland County at the city’s expense also prompted Republican outcry. Representative Mike Lawler, for example, sought campaign contributions over email to stop Democrats from “busing unvetted, adult male migrants to the Hudson Valley and dumping them here with no plan whatsoever.”

It also appeared to place some Democrats on the defensive.

In an interview, Mondaire Jones, a former Democratic congressman seeking to challenge Mr. Lawler, also criticized Mr. Adams for trying to bus migrants to other counties without the consent of local officials, saying the mayor “had handled this in a disastrous way.”

But, he said, the brunt of the blame was on Republican governors sending migrants to New York and on Republicans seeking to stoke racial fears: “They’re not serious about solving this crisis because they are using it to their political advantage. It is a very sinister, inhumane approach to politics.”

Cognizant of the thorny politics, Ms. Hochul has emphasized the need for a collaborative approach, and has pitched the relocation of migrants as a potential boon for struggling small towns losing population and farmers looking for laborers. She has also pointed to cities such as Albany and Rochester, both Democratic strongholds, that have agreed to house migrants in hotels paid for by New York City.

“I’ve been in contact with the governor a lot, as well as other members of her cabinet and staff,” said Mark Poloncarz, the Democratic executive of Erie County, which has taken about 500 migrants since June. “I believe they are taking a serious involvement.”

Even so, the pace of relocating migrants outside the city has been sluggish, and Ms. Hochul herself has noted that many want to remain in New York City. Just over 2,000 migrants have been housed upstate by the city, according to City Hall.

More than 58,000 asylum seekers are still in city homeless shelters, which has pushed the city’s total shelter population to over 110,000 — a record.

And the number of migrants arriving has shown no signs of abating: In the past week alone, more than 2,700 asylum seekers arrived in the city, officials said.

“It’s just a question of when this all sort of explodes,” Charlie King, a Democratic political consultant and former lieutenant governor candidate, said. “I just don’t know how long you can evade this, and I don’t know that it gets better in the next 18 to 24 months.”

Jay Root and Nicholas Fandos contributed reporting.