Ohio does a convoluted tax calculation for non-residents/part year residents. It calculates tax on total income, then it calculates a non resident/part year resident credit, which it subtracts from the tax it calculated on the total income. The credit is calculated as your non-Ohio income divided by Total adjusted Income multiplied by the total tax. TurboTax (TT)   does this by allocating your income as either Ohio or non-Ohio. W-2 income will be allocated by the state name abbreviation shown in box 15 of your W-2. TT will ask you, item by item, in the state section, how much of your other income is Ohio or non-Ohio income. Make sure that your non-Ohio wages show TX (Other state postal abbreviation)  in box 15 of your W-2 screen, with the TX amount in box 16.

This system allows Ohio to apply their highest tax rate, based on your total income, while only taxing your Ohio income.

Ohio has a nonresident credit allocation form.. IT NRC

http://www.tax.ohio.gov/portals/0/forms/ohio_individual/individual/2017/PIT_ITNRC.pdf 

You do not have to file a tax return for the City (local) for which tax was withheld in box 19 of your W-2. You usually cannot get a refund. Some cities have an occasional entrant  exemption for as much 20 days. You would have to contact the city to see if you qualify for a refund (although I would think your employer would know that).