When creating a new account, Game Center/Play Services will automatically be added to your account if you allow it
Go to the Settings screen, there is an "Account" category. If there exists an "Connect"-button in this category, press this button to start connecting your Game Center/Play Services account. If no such button exists, then it means that your account is connected to your currently signed in Game Center/Play Services account.
Google Play Services Only
If you already have connected to another account, then you can disconnect your currently connected account. Press the "Disconnect"-button to do so. Please be careful with disconnecting your Play Services account, since you will no longer be able to login with your Play Services account on this particular Fun Run Arena account.