Thanks to Sassy Jane Genealogy, I found out about a very nice database. The Watson House Digital Archives has information on Irish immigrant girls who arrived in New York.
The collection includes fully searchable records of the Irish Mission at Watson House 1883-1954.
Another way to find out more about Irish immigrant girls in New York is to consult Olive Tree Genealogy's New York Almshouse records. In particular the following databases may be of interest:
New York Almshouse Records 1782-1813. Records contain name of ancestor,
date admitted, age, where from or born, complaint [illness], discharged,
died, remarks. Start with New York Almshouse 1782-1813 Surnames "A" | Surnames B | Surnames C. Other surnames to follow
Almshouse Records New York 1819-1840
Almshouse Records New York City 1855-1858