How to appeal a denied refund for an unauthorized app purchase by a minor?

Here's how to request an app or media refund from Apple (From: Request a refund for apps or content that you bought from Apple - Apple Support - see the article for more detail.) If you bought this in the EU from a third party, see: About purchases not billed through the App Store or Apple In-App Purchase - Apple Support instead.

1. Sign in to https://reportaproblem.apple.com

2. Tap or click "I'd like to," then choose "Request a refund."

3. Choose the reason why you want a refund, then choose Next.

4. Choose the app, subscription, or other item, then choose Submit.

If you were charged for a subscription that you no longer want, you can also cancel the subscription using the directions in this article --> If you want to cancel a subscription from Apple - Apple Support

The above will have to be done using the account that did the purchasing. You have not told us if the child made this purchase using the child's account, or if this was done using your device.

Note that refunds are exceptions so there is no guarantee Apple will issue you with one. Apple’s position appears to be that people need to be proactive about purchases, canceling subscriptions at least 24 hours before trials end, and ensuring that others are not able to make unsanctioned purchases on a user’s account. Note that if you allow—intentionally or not—another person access to purchasing on a device, or through Family Sharing if you are the Organizer, you are taking responsibility for things they do. Apple offers these features to help you avoid this situation:

On your device you can use Screen Time to prevent unintentional or unauthorized purchases from the App Store, in-app, and other Apple services. —> Use Screen Time to turn off in-app purchases on your iPhone or iPad - Apple Support

Also on your device, require a password for App Store and iTunes purchases —> Require a password for purchases in the App Store and other Apple services - Apple Support

Use Screen Time as part of parental controls on your child's iPhone, iPad, and iPod touch —> Use parental controls to manage your child's iPhone or iPad - Apple Support

If a child made this purchase on their device as part of Family Sharing, consider activating Ask To Buy—> Approve what kids buy with Ask to Buy - Apple Support

Use Guided Access to permit a person to only use a single app —>

Lock iPhone to one app with Guided Access - Apple Support

This is a user forum, not a chat service. You can request a refund at reportaproblem.apple.com. However, you are responsible for all activity on your account.