There's no "credit" that you will claim on the New York return itself. Rather, in the New York interview you will be asked if you were a resident of New York City. When you say "no", the NYC tax withholdings are counted as part of your NYS payments, but NYS will not calculate NYC tax on the income, so it should increase your New York refund by the amount of NYC tax withheld. (By the way, make sure your employer knows you are no longer a NYC resident to stop withholding this tax).
This process could look very invisible in TurboTax, because it may be pre-calculated. But if you put in some fake entry that you were a New York City resident, you will see your refund decrease because now NYC tax is getting calculated.
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