It depends on which state you are in and whether they withheld city or school district (SD) tax.
If Ohio, your employer has probably done it right. They are required to withhold city tax, for the city you work in. You can not get that refunded. They are not required to withhold tax for the city or SD that you live in, although many will voluntarily do that.
It would be unusual for them to have withheld the wrong SD tax (but not impossible).
You can get a credit from your residence city, for tax paid to the work city. You cannot get a credit on a SD return for city tax paid (withheld) by the workplace city.