If a background check was required, noted, documneted, the job began, employement was verified,paid, fired due to background is this cause?

Your question has no simple answer. Although New York City's Laws are more accommodating after Ban the Box passed, felons still have some hurdles to jump through. And one of them requires a detailed analysis of the class of felony and the specific category as it directly relates or does not relate to the specific duties of the job being performed. You should read the following link and consider retaining a lawyer if the felony is likely to prevent you from getting similar jobs. The employer can change their mind. For example, let's say your felony conviction disqualifies you from insurance on the job, that might be a legitimate reason not to hire you. It's complicated. You need to pay for some legal consultations to get an answer you can rely on. Here's a start: https://www.nycbar.org/get-legal-help/article/criminal-law/how-to-seal-criminal-records/ If your felony looks like it is related to the work you want to perform (for example crimes of dishonesty like bouncing checks where you now want to handle cash) then you probably need a criminal defense lawyer more than an employment lawyer.