No. They will, however, upon request issue a letter, referencing the particular certificate stating they can not make any changes. The letter is issued by a Mr. Kenneth Cobb. Mention you are seeking Italian Dual citizenship. They are aware of the issues.
"Just in case" I had the letters translated and obtained an Apostille. At my appointment the errors were either not noticed or overlooked. The Consular officer did not bring them up and they remained in my bag
Some people have been successful in getting NYC Vital records to make amendments to documents that are on file in the archives. This does require a personal trip to Vital records and if you are interested in pursuing this I will try and locate the post and get you the contact name of who to deal with at Vital records.
I would greatly appreciate the contact name. I had my first appointment at the NYC Consulate today and got beaten up pretty bad, even though I was quite prepared. My father's 1908 birth certificate was "perfect" except the date of birth was 11 days late. Archives gave me a letter that they wouldn't amend the document, but I desperately need that other part where they say what is should be. I have plenty of documentation to back it up. This document will be key in correcting another document which list's my grandfather's name as Alfred, the name he used in American rather than Saverio, the Italian name of his birth. Thanks!
As mentioned, I did not have any Municipal Archive issued documents amended and the letter issued by the Municipal Archives was, in my case, sufficient.
I did have documents (BC & DC) amended at the Office of Vital Records, 125 Worth Street, Room 144 ... Room 144 is the Corrections unit, it's not the general room to obtain birth and death certificates. The person in charge of BC amendments is Katherine Lee.
Separately, you mention your father's BC was 11 days late. Late(r) from what. Usually the BC is the main document from which all other documents are viewed.