Is there a way of deleting...

OK, then it only appears if you have a subscription. When deleting purchased content there should be a confirmation dialog regarding the deletion, and then a second one asking if you want to hide from your purchase history. Say no, close iTunes and reopen, and the purchase should be listed in the cloud.

What is the issue you're trying to resolve? If you are running short of space perhaps you should move your library to an external drive.

tt2

For purchased songs you should be able to right-click and use remove download from the context menu. If you have either Apple Music or iTunes Match this will also work with media that has been matched, uploaded, or comes from Apple Music. Removing download leaves the item in your library but connected to the cloud rather than stored locally.

tt2

Ah, you can choose what you sync to your devices so as to avoid hearing stuff you don't want by syncing with selected playlists.

In iTunes under Edit > Preferences > General there should be an option to Show List view check boxes. Unticked tracks are generally excluded from syncing, and will be skipped during normal track to track and shuffle play. There is also an option in the Song Info dialog box > Options tab to skip when shuffling which lets you play the songs normally in sequence when you want to, but stops them from popping up when you shuffle.

Your device will show all of your unhidden purchase history as well as any content synced to it, so if you cannot take control by changing the checked or synced when shuffling status then hiding the purchases may actually be the best approach. On an iOS device there should also be a subsection of the library for downloaded music, which might let you filter out content you haven't explicitly added to it. I generally ask Siri to "Play Downloaded Music" because I don't have unlimited cellular data.

tt2

You're welcome. 🙂

tt2