This error means your child's phone can't reach Google's servers. Follow these steps to fix it:
On your child's device, slide your finger down from the top of the screen twice.
Make sure Mobile Data is enabled and Wi-Fi is on if possible.
Check whether you've blocked the App Stores & System Updates category, Gmail, or Google Workspace in your screen time rules and allow them if needed.
Restart the device.
Try to sign into Google again.
If that doesn't work, let's see if your kid's phone can connect to Google over Mobile Data only.
Swipe down from the top of the screen.
Turn off Wi-Fi.
Leave Mobile Data turned on.
Try to sign into Google again.
If your kid's phone can sign into Google over Mobile Data but not Wi-Fi, that means you have some third party web filter on your home Wi-Fi that is blocking it.
Log into your router settings from a parent device.
Find the web filtering, parental controls, URL whitelisting, or access control settings (the name varies by router).
Allow google.com in your filter settings.