Ohio has a reciprocal agreement, on wages, with all neighboring states. No Ohio state taxes are withheld or due and you do not need to file an OH return. But your home state will tax you on your Ohio income.
If Ohio taxes are mistakenly withheld, you have to file an Ohio return to get a refund.* You should ask your employer to stop withholding so you don't have to file an Ohio return every year.
However the reciprocity agreement does not apply to local city taxes. You are not required to file a local city return, but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount. If you are subject to a KY county or city tax, you should check to see if the KY city will give you a credit for the OH city tax withheld. KY does allow an itemized deduction (Schedule A) for local tax paid.
*If you indicated you were not a full year Ohio, at the Ohio residency screen, you will be asked if you had Ohio income and were never an Ohio resident. If you check that box, you’ll get “Ohio Compensation deduction”, under miscellaneous, at the “Here’s the Income Ohio handles differently” screen.
If you indicated you were a KY resident in the personal info section and an Ohio non resident, at the Ohio residency screen in the OH state section; you’ll get “Neighboring State Compensation” screen early in the Ohio program. There you enter the box 16 amount from your W-2 that Ohio taxed.
Or
After indication you were a non resident of Ohio and a KY resident at the Confirm Ohio residency screen
-"Here's the Income that Ohio Handles Differently"
-Miscellaneous
-Wages of Full year residents of neighboring states
Screen shot at https://ttlc.intuit.com/questions/4256143-can-not-figure-out-adjustment-allocation-part-in-my-taxes-...
All of your income is to be allocated to non-Ohio. Since some of your W-2 income has been allocated to OH, you need to re-enter that in the box titled “Compensation Earned in Ohio”. That box is specifically for the wage/salary income of full year residents of reciprocal (neighboring) states.
Screen shot at https://ttlc.intuit.com/community/state-taxes/discussion/re-ohio-credits/01/2867672#M137181
