I have an old account in the game I downloaded again. But I want to delete that account.

Sorry for the late response. Try this:

1. Open settings at the game
2. Click "Disconnect" to unbind your Game Center account

If the game is saving game data via iCloud, the data might have not been deleted when you removed the app from your iOS device.

Go to Settings > General > iPhone Storage > Scroll down and find the game and click on it

4.Delete the game

5.Install the game again from App Store and agree to login to Game Center, so your new progress will be automatically saved.

Hope this helps.

You will have to visit the specific games website and find assistance there. Apologies that is all I can tell you as I am not sure what game or if it is connected to a service such as facebook or game center.