Gamecentre

Salutations Eyepatch018, 

Game Center details are accessed in Settings > Game Center.Use Game Center to play with friends - Apple Support 

From your screenshot, it looks like your device is connected to Wi-Fi. Does the same thing happen when using cellular data? Another thing to check is that if there is VPN or security software on your device remove it, restart your device and test again. 

Here are some additional steps to take If you can't connect to the App Store, iTunes Store, or other Apple services

1. Check the system status page

• Check Apple’s System Status page to see if there’s an outage for a service in your country or region.

2. Make sure that your device is connected to the internet

• Use any web browser to connect to any website. If nothing loads, use another device on the same network to see if it can connect to any website. If no other devices can access the internet, turn off your Wi-Fi router, then turn it on again to reset it. If you still can’t connect to your Wi-Fi network, contact your internet provider for more help. If only your first device can’t connect to the internet, get help with Wi-Fi on your iPhone, iPad, or iPod touch, Apple Watch, Apple TV, or Mac.

• If you use cellular service to connect to the internet on your iPhone or iPad, make sure that you have cellular data turned on for the app that you're using. To do this, open the Settings app, tap Cellular, then turn on Cellular Data.

• Update your software

Update to the latest version of iOS or iPadOS, macOS, tvOS, watchOS, or software for your non-Apple device. Then try to connect again.

3. Check the date and time

• Make sure that the date and time on your device are set correctly for your time zone.

• On your iPhone, iPad, iPod touch: Open the Settings app. Tap General, then tap Date & Time. Get more help with date and time.

Thanks.