The company I work for is located in Atlanta,Ga. I moved to Owensboro, Ky in August 2023. I now pay Kentucky state income tax. Do I have to pay city income tax?

Usually yes, if you either live or work (or both) in a KY city (or county) with an income tax.  You may need to contact the Owensboro finance dept. for clarification.  It's not clear whether they tax workers or only the self employed. They call it an Occupational License Fee.  https://owensboro.org/page/finance-department

If you work in a different city/county than you live in, you pay tax to both and your residence city/county gives you a credit, sometimes only a partial credit,  for the tax paid to the work location city. 

The way it usually works: your employer withholds tax for the work city/county. So, you don't have to file a local tax return for the work city/county. You only file a tax return for the residence city/county.  Since you have an out of state employer, they may not be willing to withhold local tax. In that case, you may need to make estimated tax payments to your cities/counties. Contact your local city/county for forms.