tl;dr You might be fine. It depends how much total was withheld for state income tax.
This answer has a few NY specifics but is true of some other states as well.
The city tax is calculated and included with your state tax. For example, if you did not live in NYC or Yonkers, your state income tax was X. You live in NYC, so your NYC tax is Y (line 47 and 47a on your IT-1). Now the total amount you pay to the state is X+Y. The state will funnel the Y to NYC. It is possible that your employer did withhold the correct amount, and it just came on the W2 as all state income tax, because your employer does not send two separate amounts, they just send the one amount to the state. (and one to the federal, but that is not relevant here.)
For the answers:
Z = Your total income tax to be payed to the state is X+Y = Z
W = The amount that has been withheld in state income tax on your W2
O The amount that you still owe the state. It equals Z-W
- It depends. You can have a penalty for underpayment, but you will not have a penalty if any of the following are true:
a. Your 2019 W >= your 2018 Z (a few exceptions if you have relatively high income, then W has to be >= 2018 Z * 1.1)
b. If O is under a certain amount or certain percentage of Z
Unless you can tell that you vastly underpaid your total state tax (including the city tax) I would leave it alone. Note I am a guy on the internet and not an accountant or a lawyer.
- You make one payment to the state. So however you file, if you owe (Z-W > 0) you send the money to the state revenue authority.
You might be fine.
1. The above answer to 1 is similar for federal taxes. Penalty not owed if your withholding or estimated payments for 2019 >= your total tax for 2018, (110% if your income exceeds a certain amount.) No penalty if your withholding or estimated payments >= 90% of your total tax for this year (100% if your income exceeds a certain amount).
2. If you want to read all the gory details, Instructions (PDF) see pages 23 for local tax and 33 for penalty rules.
For 2020, to be sure, you can check your IT2104 (PDF) (the NY state equivalent of the federal W4) to see if the box "Are you a resident of NYC?" is checked yes.