On Friday, officials in New York still did not have answers to even the most basic questions about the children who had been separated from their parents at the southern border and relocated 2,000 miles away: How many were there, and where had they been placed?

Both Gov. Andrew M. Cuomo and Mayor Bill de Blasio sent letters demanding that information from the United States Department of Health and Human Services. When it was not forthcoming, Mr. Cuomo threatened the 10 agencies that had accepted the separated children, and which the state also regulates, to turn over the children’s names or lose their operating certificates.

But because they are bound by their contracts with the federal government not to disclose information about children in their care, the agencies said they could not share it, even with the state.

So faith leaders and local elected officials pieced together the numbers: About 60 children each at Catholic Guardian Services, Lutheran Social Services of New York and Abbott House, all in the Bronx, according to officials who asked not to be identified because of the sensitivity of the information. Another 19 had been at the Westchester children’s residence visited by Mr. Cuomo on Thursday. And there were 243 at Cayuga Centers in East Harlem, according to the latest figures from the mayor’s office. Officials seemed to be learning new numbers every few hours, as they tried to account for the 700 children Mr. Cuomo spoke of on Thursday.