How to make updated tax return, if system does not allow to send one more time the return or a new one with the same SSN that already has been accepted ?

If I am right you are saying that you have E filed the federal return but not NYC. IF I am correct and you are using pro basic then go back into the program and enter the NYC resident check mark. Now hit E File command. It will show that the federal has already been efiled but not NYC and it should read NYC ready. Hit the Efile command for NYC and away you go, mission accomplished.

If this is an individual NY resident return and it has been efiled and accepted, you will need to file a superseding or amended return.

Superseding returns can only be filed electronically at the federal level.  You'll find this note under Miscellaneous Forms > Amended Return (1040-X)/Superseded, just below the section title "Superseded Return":

NOTE: A superseded return is a return filed subsequently to the orginally-filed return and filed within the filing period. The program supports electronic filing of superseded Federal returns only. A 1040X must be transmitted with a superseded return.

If you'd like to file a superseding state return, you will need to do so on paper.  You may like to consider filing an amended NY return instead before the deadline to make the correction should you like to avoid the paper route.

The Supreme Court had ruled in Haggar Co. v. Helvering, 308 U.S. 389 (1940), as was acquiesced in GCM 36932, that a timely filed amended return is to be treated as the original return or as a part of the original return.

Many states can be filed after the federal. https://proconnect.intuit.com/support/en-us/help-article/electronic-filing/state-e-file-methods-standalone-vs-piggyback/L11dxp3xr_US_en_US?uid=lf6xr8iz

City, Municipal, etc can not. I think your only solution is to file an amended NYS with the city return attached. OR just mail the city return separately. I don't do NY so I do not have an definitive answer, just some thoughts.