“I didn’t go through everything for that,” Yoselin said, “because I had no idea that due to my domestic violence I could get a visa.” She asked to be identified only by her first name because she fears retribution from her ex-boyfriend.

The legislation granting visas to victims of certain crimes has been in place since 2000, but the agencies that can certify their eligibility vary by county and state. For a victim to qualify, certifying agencies must confirm that the criminal act is on a prescribed list and that the victim was helpful in assisting the police. If approved by the United States Citizenship and Immigration Services, the visas permit crime victims to stay and work in the country up to four years, and to apply for permanent residency.

“This isn’t some backdoor, it isn’t a loophole, it is one of the few forms of humanitarian relief that Congress has put out there to help victims of serious crimes,” said Theodor S. Liebmann, a professor of immigration law at Hofstra University.

In his State of the State address, Mr. Cuomo announced that the Division of Human Rights and the State Police would certify visas for certain crimes, joining the State Labor Department. Mr. de Blasio then announced he was expanding the city’s visa certification policy to include the city’s Commission on Human Rights.

The New York Police Department said its Domestic Violence Unit certified 152 U visas in 2015 out of 580 applications. Amid pressure from advocates, the department proposed rule changes last summer that would establish clearer protocols for its certification process.