It depends. There really are two situations here, city and state. Your primary question is about city taxes, so I'll answer that first.
Ohio has a separate city return. If your Ohio locality provides credit for the Kentucky local taxes paid, you will enter it on the local return. It is not claimed on the state return.
However, since Ohio and Kentucky have a reciprocal agreement for state taxes, you will file a Kentucky nonresident reciprocal return if your Kentucky job took out Kentucky taxes. You will get all of these back, but then you will have to pay state tax to Ohio on the same income. You continue to file an Ohio resident return for your state taxes.
Here is an FAQ on reciprocal agreements: https://ttlc.intuit.com/replies/4776822
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