Thanks for reaching out about this here in the Community.
What iOS version are you on? If there are any updates available, it'd be great for you to check them: having Spotify and iOS both on the latest versions leads to the best experience. A clean reinstall of the app would also be great for you to try, as it could do the trick if this issue is just a small hiccup.
The playlist you're mentioning, is it a public one? Or private/collaborative? Regarding the song, mind also sharing the exact link associated with it (as you're listening to it and trying to add it to the playlist)? We'll test how/if we can add the song to a playlist.
Let us know how you get on. Cheers!
Help others find this answer and click "Accept as Solution".
If you appreciate an answer, maybe give it a Like.