cannot cancel subscription purchased by child - no acces to the phone.

Correction: You must have access to the account used to purchase and item, not a particular device.

Note that Apple takes privacy and account security seriously.

Purchases must be canceled by the account making the purchase: If you want to cancel a subscription from Apple - Apple Support and Request a refund for apps or content that you bought from Apple - Apple Support

You are correct that if this is a child's account you cannot delete the child from Family Sharing. Your options are to either transfer the child to another family group or request deletion of the child's account (and everything used with that account will be lost).

Move a child to another group using Family Sharing - Move a child to another group using Family Sharing - Apple Support - "While you can't remove a child under 13 years old from your family group, you can move them to another group."

The alternative is to delete the child's account altogether which has the consequence that anything purchased with that account will not be available for redownloading or updating in the future. One way to delete the account would be to log into the unwanted "child" account and request deletion. You should have the ID and password from when you first set up the account, or can recover the password according to information provided in the referenced links below.

Create an Apple ID for your child - Create an Apple ID for your child - Apple Support

- "If you need to see the options you have for managing your child's account, you can sign in with their Apple ID and password at https://privacy.apple.com/account."

How to delete your Apple ID account - How to delete your Apple ID account - Apple Support

- Can I delete my child’s account?

"Your child can also request deletion of their Apple ID, which requires your approval."

You could also contact Apple about this request: Click here --> Choose your country or region - Official Apple Support Select your country, then a product and work your way down the menu to the desired issue. If you don't see one then try a different product until you reach a telephone or chat help request and ask to be redirected.

As for a refund for the purchase made, you will have to resolve that with Apple. The people on this forum are just users like you and there is nothing we can do other than advise you. If you have feedback for Apple: Click on this link for Apple products feedback links --> Product Feedback - Apple

Apple may not respond but your comment will be read.

See if you can set this up to prevent future purchases: Approve what kids buy with Ask to Buy - Approve what kids buy with Ask to Buy - Apple Support