Can I get reimbursed for some purchases my child did

All purchases are considered final, but you can try contacting iTunes support and explain what happened and see if they will refund or credit you : http://www.apple.com/support/itunes/contact/ - click on Contact iTunes Store Support on the right-hand side of the page, then Purchases, Billing & Redemption

To try and stop it happening again you should be able to turn off in-app purchases via Settings > General > Restrictions > In-App Purchases 'off', and you can set 'Require Password' to 'Immediately' on the Restrictions screen (the default is a 15 minute period during which it doesn't need to be re-entered).