How do I delete a playlist from a synced ipod but not remove it from the library in itunes?

With the device connected select the music tab for it and make sure that you're using the option to sync with selected playlists etc. Make sure the lists you want to exclude are not selected, then sync to update the device.

tt2

Select the iPod in iTunes (click its device button on horizontal bar) to show the its settings screen in iTunes window. Along left side (in the sidebar), under Settings, click Music. To the right, the iPod's Music setting screen is shown. This is where you set up automatic syncing (with the Sync Music checkbox). There are two options for how iTunes syncs the iPod. One is to sync Entire music library. That is probably the currently selected choice, if "everything" is being synced. The other option is to sync Selected playlists, artists, albums, and genres. Use that choice.

However, this may not work conveniently unless you do some prep work in your iTunes library first. A convenient way to do what you describe is to create a new playlist in your iTunes library. You can call it Child's Songs (replace Child with actual name 🙂). Add all of the songs you want on the iPod to that playlist, leaving out songs you do not want on the iPod. If you have more than one child, you can create a different playlist for each child (or use the same playlist). Back on the iPod's Music settings screen, under Playlists, find and select (checkmark) that playlist. You don't need to select anything else on that screen. Click Apply to sync your selection to the iPod.

Going forward, to update songs on the iPod, update the Child's Songs playlist in your iTunes library (add/remove songs). The iPod does not need to be connected. The next time you connect the iPod (or click Sync if already connected), iTunes automatically updates the iPod with the same changes.