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Dear Ms. Caputo,
Thank you for contacting us regarding the Hebrew Orphan Asylum records that were presented by Ancestry and Family Search online genealogy. The Orphanage was not US Government funded, which means that we do not hold any records of the institutional activities. I googled the institution and came up with a wikipedia.org result that posts archives resources. Here is the statement that may be of help to you:
“The archives of the institution are on deposit with the American Jewish Historical Society, at New York's Center for Jewish History. The admission and discharge registers for 1860 through 1928 can be searched via the web.
Nearly fifty oral history interviews about the institution are on deposit at the New York Public Library as part of the American Jewish Committee Oral History Collection.”
I would suggest you indulge in extra time with the New York Public Library as mentioned as well as the New York State Archives, both of which may be of much help in records or recommendations. There were very many resources listed in the New York Public Library catalog especially.
I hope my reply is helpful. Best of luck with your genealogy research!