Game Center is no play no longer offered as an app in iOS 10. You sign into Game Crnter in Settings.
You may be able to get your progress back if you restore a backup, if you have backup that contains the game data. Farm Ville may backup game data with their own servers but you would have to check with their app support in order to determine that.
Bear in mind that if you restore a backup, you restore everything in that backup going back to the date and time of the backup. Everything else that you have done of the device from that day forward, will be erased.
Restore your iPhone, iPad, or iPod touch from a backup - Apple Support