If you're having trouble sending and/or receiving mail on Android using a Google supervisied account or a Google child account, check "Parental settings > Filters on Google Chrome". If it's on "Only allow certain sites", change it to "Try to block mature sites" and this should resolve your issue.

Another thing to look out for is the 15 GB storage quota. Your kid account can easily reach that if you use Google Photos in original quality and once reached, Gmail won't receive or send email anymore.