Hi dsafian,
We'd like to see how we can help out, but we first want to make sure we understand exactly what is happening.
Both the Mac and iPhone have Handoff disabled.
While working on your Mac, you are listening to a podcast on your iPhone.
If you happen to send a document to the trash, on your Mac, you say the Podcast app shuts down on your iPhone.
Is that correct?
To begin, since the app is closing on your iPhone, see this article and follow the steps to see if this helps with the app closing. If an app on your iPhone or iPad stops responding, closes unexpectedly, or won't open
If you still have the same issue, we recommend reaching out to Apple Support by going to Get Support.
Have a great day!