I need oftentimes to move files (TV recordings) from my computers to kid's computer.
The destination folder is on the kids' computer USB HD (too many files and too big for storage on the small main HD).
Kids have an account there which is used only for watching these TV recordings (no games, no Internet on that account and they have no access to the files from other accounts).
Guessing a bit here because I can't check it locally, but it should work in theory.
I assume you have an Admin account on the kid's Mac?
Log in to that, got to System Preferences > Sharing and enable file sharing (tick/check the box).
Log out and do the same on your Mac. With both Macs connected to the network, you should now have the kid's Mac in the Finder sidebar.
Log in to your account on the kid's computer by highling the comp in the sidebar and choose connect as; then enter your short name and password for the remote account.
You should now see the ext HD below the Mac in the sidebar.
Drag and drop your files to there.
When you're finished, go back and turn off file sharing on the kid's Mac.