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In the latest version of the Spotify phone app, you should be able to fix it.
1. Search for the affected arist's page on the Spotify phone app. You will need to go to their artist page.
2. Ordinarily there should be a Follow button up the top right near the artist name, there is like a 🚫 symbol. Click on that.
3. You will need to restart the Spotify app on your desktop.
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You can try adding a song to your library, it worked several times for me.
I contacted Spotify support on Twitter @spotifycares and they were able to fix the issue by logging in to my account and unblocking those artists. Apparently, that's the only way to fix this and there is no option in the interface for it.
Worked like a charm. Thanks.
I have logged into the app on my mobile. This solution does not work. There is not such a symbol or Follow button up.
Click on the three dots on the top right corner in the artist's page, you'll then see the "Allow to play this" option.