There is something you can try using familysearch.org. If not a member, join, it's free.
Do a new search, enter the last name in the surname field, and ONLY the FIRST Names of both parents in the mother and father field (you can try one parent name also, but the list will be longer). Under TYPE, check off BIRTHS. In the birth location field, fill in New York City or other part of NYC and do a search. Change location and keep trying - there are never guarantees.
You'll get a list of all the birth records that got indexed for the surname and the first names of the parents unless the name was butchered badly on the records. Try variations
If you are lucky, you'll find a couple of promising leads. Some of the data on the BC's are extracted. With the film number and certificate number, you can go to the Family History Center when they reopen and find the images on line. I have found several children that were unknown in our family in this manner and had died at birth or even within four or five years later.