My son has a child account under my iTunes and when I approve a purchase for him it gets charged to my credit card, even though I have an iTunes balance credit from gift cards. Is there a way to circumvent this, because he added his Christmas iTunes gift card under my iTunes, but we can't seem to use it for his purchases.
If you add any iTunes gift cards to his Apple ID (not yours) any purchases he makes (even via your approval) will first come off his balance, not yours.
If/when his balance runs out, then any purchases would be charged to your account balance. If you have no credit on your account, then it would get charged to your credit card (assuming you have entered one).
Perhaps you buy him a new gift card (as payment for the one he added to your account) and add the new one to his account. That should solve the problem.