The Cayuga welfare agency has received the majority of the children separated from their parents at the border, said Bitta Mostofi, the city’s commissioner of immigrant affairs. She said that Cayuga currently serves 600 unaccompanied minors (including those who did not come with parents), and that it planned to add 300 more spaces over the summer.

Cayuga Centers, which has several locations, did not return repeated requests for comment.

The children have been separated from their parents at the southern border as part of the federal government’s zero tolerance policy, in which adults were being prosecuted for entering the country illegally, and their children were taken from them and placed into custody separately. More than 2,300 children have been taken from their parents in the last two months.

But it was unclear how Mr. Trump’s executive order ending the separation policy would affect the chaotic situation for the children in New York.

José Xavier Orochena was outside Cayuga on Wednesday. He is the lawyer for Yeni Gonzalez, an immigrant from Guatemala whose three children are being held at one of their centers, he said.

Ms. Gonzalez fled gang violence in Guatemala with her three children, Mr. Orochena said. After they arrived in McAllen, Tex., over a month ago, the children were taken from her and sent to New York, Mr. Orochena said. She is being held in Eloy Detention Center in Arizona. Mr. Orochena said that the children and their mother have communicated by phone.