Q: In app purchases by minor child

It happens.  You should review

iOS Restrictions:

http://support.apple.com/kb/HT4213

iTunes: Parental Controls:

http://support.apple.com/kb/HT1904

Mac App Store Parental Controls:

http://support.apple.com/kb/PH11496

Mac OS X: Parental Controls (video)

http://support.apple.com/kb/VI28

Regarding the purchases:

To report a problem less than 90 days:

http://reportaproblem.apple.com

Sign in with the AppleID and Password used for the purchase.

How to report an issue with your iTunes Store, App Store, Mac App Store, or iBooks Store purchase

http://support.apple.com/kb/HT1933

To report a problem greater than 90 days:

http://www.apple.com/support/itunes/store