Hey there Chrisanngurley,
Thanks for reaching out to Apple Support Communities. We understand that you're having some issues connecting to your Game Center. We're happy to provide some assistance.
This may seem like a simple step, but have you restarted your device? Restarting your device can resolve unexpected issues. Take a look here on how to restart your iPhone: Restart your iPhone
If the issue persists, try a different connection such as a different Wi-Fi network or using a cellular connection.
If the issue persists after that, see if the issue occurs on another device to help isolate the problem.
We hope this helps.
Have a great day!