Good afternoon AnotherPerson21,
Welcome to the Apple Support Communities!
I understand you are getting a message to update your Game Center account. Since your iPad mini can't be updated past iOS 9.3.5, it's not asking for a software update. It's most likely asking about your Apple ID. Tap Settings > Game Center to confirm you are using the correct Apple ID and Game Center Profile. For in-app purchases with games, you may want to check to make sure your billing account is updated. You may need to enter your credit card expiration date and security code again, and then tap "Done" to update it.
Change or remove your Apple ID payment information