Q: How do you remove an app? In this case, it was viewed, purchased, free in this case,  downloaded, then removed, but still appears in the purchased section, I do not want the younger kids to see it? There appears to be no remove or hide function?

You cannot edit that information since it is really just a receipt for your account. Why would your kids be inspecting your account's purchase record?

You can delete the app from iTunes, but that will also delete it permanently from your iDevice once you sync it.

You can hide iTunes purchases:

iTunes Store: Hiding and unhiding purchases - Support - Apple

If the OP's kids are prying into the purchases history, they may quickly figure out how to unhide purchases, as well. I would wonder if this would work in the long-term given the kids' curiosity.

Maybe I misunderstood. I assumed the OP meant the Purchased section in the App Store app.

In that case they would still need the Apple ID password to access the account and unhide the purchases.