There are a few things that you can check to see if it brings your Game Center connection back to normal.
First, enter your device’s settings and scroll down to tap on your General settings tab. Here, look for Restrictions and enter your Restrictions passcode. Then, scroll all the way to the bottom of the page to the Game Center restrictions to ensure that both "Multiplayer games" and "Adding friends" is on.
Secondly, go all the way back to the beginning page of your Settings and scroll down to your Game Center tab. Here, make sure that the "Game Invites" are enabled. And that you are signed in as yourself in the "Game Center Profile". As for the Friend Recommendation section, it is a personal preference so you can choose whether you want to enable "contacts" or "Facebook".
This should help!
If you’re still having trouble connecting your device to Game Center, please check out our Tutorial Video section, maybe this can help you out.