As you may or may not know, music originating from Apple Music cannot be added to an iPod Classic. (It can be added to an iPod Touch, but not the Classic.)
So if you have an Apple Music subscription, perhaps what has happened is that you have bought the song but either:
- it has not been downloaded to your iTunes Library, so the copy that you are trying to add to the Classic is in fact from Apple Music or
- you have downloaded the purchased copy of the song, but you have selected the Apple Music version to copy to your iPod rather than your purchased copy
There may be another scenario, but I haven't thought of it yet.
One thing to do is make sure you can find the purchased copy in your iTunes Library and that it can be played in the library. If you select a song (in the library) and tap the Edit/Song Info/File, you will see whether the copy you're looking at is a purchased copy or an Apple Music one. Notice that for the song I'm looking at, the kind is a Protected AAC audio file, but also notice that the purchase date was on 10th March 2009 (UK date format). The relevance of this is that Apple no longer sell music in protected format. I just didn't make any effort to replace this song with an unprotected one*:

In the second screenshot, notice that the purchase date was 7th July 2009, but that it is not protected. The kind states Purchased AAC audio file:

I'm not sure exactly what information you will be able to find for an Apple Music song since it's not a service I use.
It is possible that the Karima Francis song can be added to an iPod Classic, even though it's protected, but I don't recall whether it was ever on my now non-working Classic. The song was actually an Apple giveaway, which may be why it is protected.*
* Post updated by author after intial posting.