I'm following up on the above. Without some further information from you as above, I am limited in what I can say on the matter but in the hope it is helpful nonetheless, I will provide you with the following broad answer. If you are able to kindly provide me with the above further information or if you have any further questions generally, I will be delighted to expand on the following - please just reply back to me in this case:
Notwithstanding the above, in general terms the position is likely to depend heavily on whether you authorised your card on an Apple account / device and then gave that device to the child (lets call them C) and C then made transactions on that device without your permission or if C used your card details without your permission and registered it on an Apple account themselves whether this was one of your devices or one of C's and then made the transactions.
In the case of the former, you would likely be largely reliant upon apples refund policy because whilst he did not authorise the transactions, you did authorise your card in the circumstances on the Apple account which would have included authority for Apple to charge transactions made using the account to the card. Accordingly, from the card issuer's point of view, they will say you authorised use of the card and in the abuse of your Apple account as a matter as between you and Apple.
The position is very different however if C used your card without your permission in the second of my two above examples. In this case, this will amount to fraudulent use of your card and is a matter which you can raise as a charge back through your credit card or debit card provider. Of course, this potentially could expose C to a criminal investigation if the card issuer chooses to pass the matter to the police and this may be a matter which she would want to consider carefully perhaps not in relation to whether to make a report not but at least as an ongoing consideration. If you are not Cs parent, then you may wish to contact their parent to seek their involvement.
I hope the above is of some assistance but if you have any further questions, please revert to me