We tried but weren't able to replicate this issue. In most cases disconnecting and reconnecting the app should do the trick. When the add to button is still not visible, this can be caused by a track not being recognized by Spotify or having multiple versions. This can also happen when a track is not available on Spotify at the time.
Does the issue persists, when you log into your account on another device? That way we can check if it's an account-related issue. It's also a good idea to try with another network.
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"Difficult roads often lead to beautiful destinations." ―Zig Ziglar
This has been a problem for years before 2017 when I posted my comment on 2017-09-0408:45 PM. Shazam had a page acknowledging the issue that was dated years before 2017 that seems to have been taken down since Apple acquired Shazam.
I've previously tried uninstalling/reinstalling both apps, upgrading my phone across multiple of the latest iOS releases (from 2014 - 2017), upgrading/changing my phone. It's clearly an issue with Shazam and how they interface with Spotify, as I've since back in 2017 switched to SoundHound and haven't had a single issue since 2017. It would be nice to have a working Shazam as it's built-in to Apple so thought I'd comment as I am confident this isn't a simple user error given I swapped everything and the only thing that worked is changing off Shazam.