You cannot get that refunded.
The reciprocity agreement, between KY and OH on state taxes, does not apply to local city and county taxes. You can not get those refunded.
But if you live in an Ohio city with a tax, they may give you a credit or partial credit. Many do. You need to check with your city.
If you are subject to Ohio School District Income Tax (SDIT), you can not get a credit, for KY local taxes withheld, on an SDIT return.
