Santos Lopez uprooted his family and walked nearly 2,000 miles on a dangerous trek from Honduras to the United States last spring to escape from a violent gang that was extorting him. The group demanded a monthly payment, he said, to allow him to run his car shop in peace.

Like many others, Mr. Lopez and his family hoped their experience would persuade their adopted country to give them asylum, which is granted to those who face a “credible fear of persecution in their country of origin.” A grant of asylum would allow them to work and eventually apply for a green card and citizenship.

But more than a year after his family — including his wife and two daughters — arrived safely at the southern border, it seems likely they missed the deadline to apply. Mr. Lopez, 42, said he was seeking help from a lawyer.

Mr. Lopez and his family are among the millions of migrants who have arrived at the southern border in the past year. Many, after telling border agents about abuse and persecution that they experienced, a first step in the long and complicated process of seeking asylum, have been temporarily released as they wait for their immigration cases to wind their way through courts.

But even as migrants have applied for asylum in record numbers, advocates and immigration attorneys say that without additional legal support, many — perhaps the majority — will miss their application deadline and fall into a more perilous category of immigrant: the undocumented.

“Our immigration system is broken,” said Henry Love, vice president for policy and advocacy at Win, which runs 14 family shelters and has a contract with New York City to house migrant families.

“You’re going to have so many people who won’t have the opportunity to apply for asylum simply because of the logistical complications of it,” he said, adding: “I have a Ph.D., and there’s no way I could do it.”

As migrants have fanned out across the United States, many leaders and immigrant advocates have begun referring to the newcomers as asylum seekers, not simply immigrants. In the fiscal year that ended in September, a record 250,000 asylum applications were filed nationwide, increasing the total number of pending applications to nearly 1.6 million, according to the Transactional Records Access Clearinghouse at Syracuse University.

New York City has become the top destination for newly arrived migrants. Between March and May of this year, nearly 39,000 new immigration court cases were filed in New York City, compared with about 11,000 in Miami-Dade County, Fla., and about 16,000 in Los Angeles County, according to the clearinghouse’s data.

As of June 25, more than 81,200 migrants had arrived in New York since last spring, and 50,000 are housed in city shelters, according to city data.

Migrants who want to apply for asylum generally have 12 months to submit their completed applications — though many, like the Lopez family, are uncertain about the timeline. If people don’t submit their applications in time, it can jeopardize the new lives they were constructing in their adopted cities.

People who formally seek asylum are allowed to remain in the country to await a decision in their case, and applicants can apply for temporary employment authorization 150 days after successfully filing their application.

Many new arrivals in New York, frustrated by the wait to get a work permit, have looked for jobs in the underground economy and joined the existing pool of undocumented workers. But without proper documentation, they remain vulnerable to deportation and exploitation.

Asylum seekers miss their application deadlines for myriad reasons.

The application itself can be incredibly difficult: It is 12 pages long, in English, and includes questions like the last five addresses where the applicant lived, the names and addresses of relatives and a portion to explain — in detail — what harm or mistreatment they have faced. Missing one question can result in the return of an application.

Just knowing where to submit the application is complicated and depends on the specific details of each new arrival’s case.

Additionally, new arrivals must quickly engage with the immigration court system to avoid being immediately deported. But they often discover that critical immigration documents have been sent to incorrect addresses, meaning they miss important court dates, putting them at risk of being deported. Some new arrivals have also reported receiving initial court hearing dates that are long past their deadline to apply for asylum.

They must do all this through a backlogged court system, which can make the process more complicated and time-consuming. And many migrants navigate the process alone because of a shortage of immigration attorneys and advocates.

Jodi Ziesemer, director of New York Legal Assistance Group’s immigrant protection unit, said that even for those who regularly practice immigration law, the administrative complexities can be confusing.

“Even when you get down to submitting the application, it can be unclear where you need to submit it for it to be accepted,” she said.

But the main persistent problem is that many asylum seekers are not aware — and often are not told by officials — that they are working against a time clock to submit their applications, advocates say.

Under normal circumstances, it can take six or seven years for cases to be adjudicated. Ms. Ziesemer said that the current stress on the already struggling immigration court system will inevitably prolong cases even further.

It may soon become “untenable” for the courts to work through the growing backlog of cases as they’re operating now, she said.

Christine C. Quinn, the president and chief executive of Win, said that staff members at her organization were surprised when they learned that many of the families in their shelters had an asylum application deadline on the horizon.

“What happens when we have tons of these shelters where we know that the overwhelming majority, probably like 95 to 98 percent, have not actually started the application process that they’re eligible for?” said Mr. Love. “All these people will time out.”

Win has partnered with legal aid attorneys and created an applications clinic. The organization’s staff identified families that were near or past their one-year deadline and provided transportation to meet with attorneys. The group was able get legal help for 25 families.

“I am sympathetic to the fact that the mayor and the city were blindsided by this,” Ms. Quinn said in May, adding, “But now, I think we really should be in a different place from where we are.”

In June, Mayor Eric Adams announced the creation of the Asylum Application Help Center which will bring together immigration legal service providers and pro bono attorneys and aims to serve thousands of asylum seekers by the end of the summer.

Attorneys at the center will help asylum seekers complete their applications and file them, but will not represent clients through the court process.

As the city works to help migrants, the flow of newcomers continues.

Ruthmary Murillo, 23, originally from Venezuela, came to the United States from Colombia last September with her husband, Diego, and their two young children. Ms. Murillo said her family plans to apply for asylum, but they haven’t started the process yet.

“When I came I didn’t know anything,” she said, adding: “In the shelter we didn’t find out about anything.”

Her family is now connected to a help center at Good Shepherd Lutheran Church, in Bay Ridge, Brooklyn, she said.

“They have helped orient us,” Ms. Murillo said. “Now we know a little bit more about what we need to do.”