Why can’t I get a refund my child made without my permission ?

Refunds have to be requested by the purchaser. You do not provide details as to who made this purchase and on which device.

See your purchase history in the App Store or iTunes Store [iOS, computer, or web] - Does not list purchases made by others under Family Sharing; those have to be checked from their accounts. View your purchase history for the App Store, iTunes Store, and other Apple media services - Apple Support

To "Request a refund for apps or content that you bought from Apple", click here → Request a refund for apps or content that you bought from Apple - Apple Support

Report a Problem - https://reportaproblem.apple.com/?s=6

If a reporting a problem request is unsuccessful you can also try clicking here → Choose your country or region - Official Apple Support Select your country, then a product.

Also refer to:

Require a password for App Store and iTunes purchases - Apple Support

1.

Refunds are not guaranteed. For more information, see the Terms & Conditions.

We can't help you, sorry, we are also users of Apple products & services like you. 

2.

To avoid the case in the future, see:

Require a password for App Store and iTunes purchases

Require a password for App Store and iTunes purchases - Apple Support

Manage App Store purchases, subscriptions, settings, and restrictions on iPhone

Manage App Store purchases, subscriptions, settings, and restrictions on iPhone – Apple Support (UK)

Prevent in-app purchases from the App Store

Prevent in-app purchases from the App Store – Apple Support (UK)