If you live In New York City or Yonkers, there is city income tax. If you live somewhere in the rest of the state, there is not. For city income tax in New York State, it does not matter where the job headquarters are or where the job is.
To find out how much tax should be withheld, go to
https://www.paycheckcity.com/calculator/salary/new-york, enter the information (type in your pay, change weekly to biweekly, etc.), and click "Calculate".