You still need to pay for the subscription that was purchased. Cancelling only prevents the subscription from being renewed, but will not make the initial charge go away. It is true you will not be able to use the App Store to download/update apps until that is resolved.
There are many methods to prevent a child from making purchases, but they must be used by you.
- Require Password for purchases
Require a password for purchases in the App Store and other Apple services - Apple Support
- Prevent in-app purchases or subscriptions
Use Screen Time to turn off in-app purchases on your iPhone or iPad - Apple Support
- Prevent downloading any apps
Use parental controls on your child's iPhone and iPad - Apple Support
You can try and request a refund for a purchase, but it would only be possible after the payment is processed and is not guaranteed. In most cases all transactions in the App Store are final, but on some occasions Apple may provide a compassionate refund, but it should never be expected.
Request a refund for apps or content that you bought from Apple - Apple Support