my child purchased something without permission. I have not been refunded

In most cases All Transactions are Final in the App Store. In some cases Apple may provide a compassionate refund and they appeared to do that on one item, but any refund should never be expected.

There is no need to remove the Ask To Buy option when granting a request for a purchase. You will be given the notification to approve each item without having to shut it off.

You can also choose to require a password for purchases where only you would be able to approve the purchase by entering a password.

Require a password for purchases in the App Store and other Apple services - Apple Support

You're lucky you got the £4.99 back.

They can't know you didn't make the purchase for your child, and then are simply trying to claw back the purchase amount.

Think about it. If everything was that easily refundable, just about every purchase would be challenged by the user as "someone else did it", or "I accidentally purchased an app". All for the purpose of trying to get something for nothing.

With Ask to Buy turned off, you will not get a Notification. You should get an email receipt, but they do not come in immediately after purchase and could even take a couple of days. The email gets sent to the mailbox of your Apple ID.