Dear Brooklyn Tenant:
When the landlord accepts rent after the expiration date of the tenancy as presented in the termination notice, and before the landlord starts the summary proceeding, the landlord creates a new month to month tenancy. Returning the check after deposited in the bank will not reverse the fact of the acceptance and creation of a new month to month tenancy.
But the landlord is allowed to accept rent for the current month, as the tenancy exists in the current month and the rent would come due on the first day of the month. If the termination notice is served before the rent due date, the acceptance of the rent as it came due does not wipe out the notice. But when the landlord takes rent for a new month after the end date of the tenancy and before the landlord starts the summary proceeding that would be a defense.
A tenant with an extant termination notice should consult with an attorney. The termination notice is a creature of statute (New York Real Property Law Section 232-a) and there are many mistakes made by a landlord when terminating a tenancy that would defeat the summary proceeding. The issue of acceptance of rent and when the acceptance occurred is one of many issues that an attorney may exploit to restore your month to month tenancy and defeat the summary holdover proceeding.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.