Hello CedBars,
Thank you for using Apple Support Communities!
Based on what you mentioned, Game Center is not working as expected due to network issues.
According to the images, it appears that you are in your Account from App Store > [your user] > Game Center. Try going directly there from Settings > Game Center to see if you can get results. Make sure your iPhone is updated to the latest version. Here's how: Update your iPhone, iPad, or iPod touch.
Then check if you can access Game Center options following the suggestions here: Use Game Center to play with friends. If possible, check if you get the same results with an alternate Wi-Fi connection. Lastly, review: Recommended settings for Wi-Fi routers and access points. This will help ensure the right ports are open. You may need to refer to your router manual.
Best wishes!