Early last week, Mayor Eric Adams called for federal assistance to help with what he said was a flood of 2,800 asylum seekers who were making it difficult for New York City to fulfill its legal obligation to provide housing to those in need, known as the right to shelter.

Mr. Adams said the influx was partly caused by migrant families “arriving on buses sent by the Texas and Arizona governments.”

The mayor’s comments created his latest political entanglement, this time against a national backdrop: His assertion began a back-and-forth with the governors of those states — much as a similar accusation from the Washington, D.C., mayor, Muriel Bowser, had drawn headlines a few days earlier.

But advocates for the homeless said that Mr. Adams’s strategy of blaming the entire spike in shelter population on migrants is a distortion that distracts from the city’s homeless crisis, leaving vulnerable families caught in the middle.

Late last week, the city acknowledged that it had violated the right-to-shelter law when four families slept overnight at a homeless intake center. The Legal Aid Society said it could not recall such a violation since at least 2014.

And on Wednesday, there were more than 48,600 people in the shelter system, compared with just under 46,000 in May.

The image of children sleeping on hard plastic chairs in an office lit by fluorescent lights harkens back decades to a grittier era of New York, an image that Mr. Adams would like to erase as he seeks to attract the Democratic National Convention in 2024. That effort may already be hampered by the perception — which he sometimes propagates — that serious crime is spinning out of control in the city.

The city’s failure to build adequate amounts of affordable housing should not be blamed on “people fleeing violence from other countries,” said Shahana Hanif, a city councilwoman who is chairwoman of the Council’s Immigration Committee.

Some facts are not in dispute. The number of migrants arriving at the southern border last month is the largest it has been in years. At the same time, it’s not unusual for the shelter census to increase this time of year as families that have been doubled up in overcrowded apartments are often asked to leave at the conclusion of the school year.

The city is also struggling with staffing issues that slow application processing for shelter placements, as well as landlords who won’t accept vouchers for permanent housing, which would free up space in temporary shelters — all while trying to build and open new housing options that often face local opposition.

Other signs of trouble were evident. The Legal Aid Society asked the Department of Social Services commissioner, Gary P. Jenkins, in an email on July 11 to explain why the vacancy rate for shelters that house families with children was below 1 percent with approximately 30 available units on July 6. The optimal goal is to keep the vacancy rate at 3 percent for greater flexibility, Legal Aid said.

Mr. Adams, at a news conference on Thursday, said the city was working hard to house homeless families, adding that it was “not acceptable” that four families slept overnight in the intake center. Legal Aid suggested that as many as 13 families may have had to sleep in the intake center last week, but city officials dispute that.

“We got housing for families,” the mayor said. “If those families were sleeping on floors for days, like were done before, I could understand the Legal Aid’s critique.”

The mayor, however, insisted that an influx of asylum seekers had compromised the city’s ability to provide adequate shelter, and anecdotal evidence seemed to somewhat support his contention.

The commissioner of immigrant affairs, Manuel Castro, said he had spoken to many asylum seekers in the Bronx, Staten Island and Midtown.

Staff attorneys with the Legal Aid Society, who often visit the intake center, said they have also noticed an increase in the number of asylum seekers, but not at the levels Mr. Adams is claiming.

One night last week, when the city had processed housing applications from 120 families, Kathryn Kliff, a Legal Aid staff attorney, said she spoke to 15 families who had arrived from Texas. The intake centers are also being filled by local families and families who traveled to New York independently, she said.

City officials began noticing an uptick in asylum seekers at the end of May and had been discussing it internally when they decided to go public with their call for federal aid last week, said Fabien Levy, a spokesman for Mr. Adams.

The number of asylum seekers, Mr. Jenkins said, was “definitely an estimate” but one based on an examination of trends and speaking with families during their housing assessments.

Catholic Charities of the Archdiocese of New York, which works to help migrants, has seen an increase in the number of asylum seekers being bused in throughout the country. Many arrive in New York City in need of basic necessities such as food, clothing, medical care, shelter and legal help. “Chaotic” is how the group’s executive director, Monsignor Kevin Sullivan, described the current state of affairs.

Alania Hughes, 21, of the Bronx, was one of the people who spent the night in an intake center last week with her 1-year old son. When she arrived at 9 p.m., the line was out the door. Her baby slept in his stroller that night. While there, she spoke to several families that had arrived from Texas.

“I just tried to stay up,” Ms. Hughes said. “I just wanted to get into the shower.”

She and her son were placed in temporary housing the next day. As the political jousting between Mr. Adams and the governors of Texas and Arizona continues, Ms. Hughes is unsure of when she will get permanent housing.

“Mayor Adams’s problem is not with Texas,” Mr. Abbott said. “It is with President Biden’s refusal to stop this border crisis and secure our southern border.”

Mr. Adams dismissed pushback from the governors of Texas and Arizona.

“Now the people who are sending people away, they tell you they did something differently and automatically you believe them,” the mayor said. “I wish you’d treat me that way.”

“They can say what they want,” the mayor said. “They were wrong. They ended up here because they didn’t get the support there.”

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