I had the same problem on both family phones, a 6+ and a 5s (although not on any of our iPads.)
My temporary work around was to try Downcast and Overcast, both great podcasting apps - and both have watch apps too!
The only thing that worked was to plug into iTunes, do a "restore" which resets to factory settings and makes a deep wipe of the phone, then start over from my last good backup. It was especially bad since I couldn't make a good backup because of the software glitch and had to use old backups. My daughter's wifi icloud backup was unusable too, and she didn't have an old iTunes backup, so she had to start over completely from scratch. This was a serious pain in the *** and took many, many hours. My young daughter was very sad. Also I lost weeks of painfully earned activity data from the watch - a serious blow!
I would say don't do this, but it's so important be be able to make good backups!!! What if I had lost a year worth of data? Mostly stuff is in iCloud but....