I recently updated to Mac OS Catalina, so I no longer have iTunes. I've already tried deleting the Podcast App and re-installing, but there are still podcast episodes "Synced from iTunes."
Here's the workaround I found - on your iPhone, launch Podcasts, select a program/show, go into Synced from iTunes and tap the plus button on each episode and then tap the download button.
Once the download completes for all the episodes, Synced from iTunes will disappear.
This workaround was based on an Apple Communities post I saw.
I also have a post about this on my website if you are interested.