If you Approve a purchase for your child, then later want to delete that app or purchase. How do you permanently delete the app? Everything I have tried, they are still able to download without getting permission again since I originally approved it.

Read this link please --> Approve what kids buy with Ask to Buy - https://support.apple.com/HT201089

I think one a purchase has been approved there are no controls on future downloads or a way to block an item.

"If a child redownloads a purchase, downloads a shared purchase, installs an update, or uses a redemption code, the family organizer won’t receive a request. "

Incidentally:

Apple products feedback links - http://www.apple.com/feedback/