Even as children were arriving, Cayuga Centers continued to advertise for more foster families and for staff, holding weekly on-the-spot job interviews. Separately, it has put out notices requesting bilingual foster families, advertising in English and Spanish the $1,000 a month per child in tax-free payments foster families could receive.

Every morning, cars roll up to its two centers, one in East Harlem and another on 125th Street near Broadway, dropping children off for day programs. At night and on weekends, the children stay with foster families, as many as eight to a house, often sleeping in bunk beds. During the day, the children can be seen near the 125th Street center, playing soccer and other games on the grounds of Manhattanville Houses.

Many of the separated children have now left Cayuga’s care. In interviews, a dozen separated children ages 5 to 17 and some parents generally praised the treatment received while in Cayuga’s care. The foster homes were mostly in the Bronx and Upper Manhattan, with Spanish-speaking foster parents, many of them Dominican.

Yolany Padilla, from Honduras, said her 6-year-old son, Jelsin, went to performances with his foster mother, a professional singer of traditional Dominican music, and danced bachata with her and the several other foster children living at her home.

But Jefferson Che Pop, also 6, from an indigenous village in Guatemala, was returned to his father, Hermelindo Che Coc, with a rash on both arms, his stomach and back, and a bruise above his eye that he said he had received when he had fallen out of bed, his father said. Mr. Che Coc said in a telephone interview that he did not blame Cayuga Centers or the foster family for his son’s physical condition. “I thought I would never see him again,” he said.

Meanwhile, the federal government has asked Cayuga to increase its capacity to 900 children in the next month, said Alphonso David, the counsel for Gov. Andrew M. Cuomo, who said all agencies were required to report an increase to the state. Cayuga had not yet done that, so it was unclear, Mr. David said, whether it would still be receiving unaccompanied children coming from the border in the coming weeks.

And even despite the millions of dollars in government grants it received to care for these children, it is still soliciting donations through an Amazon Wish List for items for the children: backpacks, coloring books, crayons, jump ropes, lunchboxes, art supplies, a portable badminton net and other sporting equipment. A link on the Cayuga website allowed the public to donate through PayPal.