I moved june 1, 2022 between two ohio cities. My employer did not change the local withholding. they say they can do nothing for 2022. is this correct?

Ohio has a complicated local tax system.   We could be more helpful if you would be specific about the cities involved.  What city did you work in?  What city did you live in until the move ?  What city did you move to?

Q.  I moved june 1, 2022 between two ohio cities. My employer did not change the local withholding. they say they can do nothing for 2022. is this correct?

A. Simple answer: Yes.

Like others have said, it depends on the details. For Ohio City tax, withholding is primarily based on where you work, not where you live.   Employers are only required to withhold for the work city, although many will voluntarily withhold for the resident city as well. If the city you work in has an income tax, your employer should continue to withhold for that city, even though your residence changed.  

You normally only have to file a return in the city you live in. It is not necessary to file one in the city you work in, if different. Your employer will remit withholding to the city you work in, but you do not file a return unless you're claiming a refund or your employer didn't withhold enough (both highly unusual). The city you live in will allow a credit or partial credit for the withheld tax you paid to the work location city.

If your employer was withholding for  your previous resident city, and withheld too much,  you will claim a refund on the part year resident city return. You do not claim a credit, for excess tax paid to the 1st resident city, on the 2nd resident city return.  Credit is only given for work city withholding, not mistaken withholding for resident cities.