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I bet it would quit playing if you just turned off the screen too. Try it, start it playing and then put the screen to sleep and wait 30 second to 1 minute. What is happening is the app playing your podcast and/or music is going to sleep in the background. If you have a Samsung, and I'm betting that you do, go to Settings>Battery and device care>Battery>Background usage limits>Sleeping apps/Deep sleeping apps, and check to see if you find your apps in either of those lists and remove it from whichever list you find it in. That should take care of your issue.
If you do not have a Samsung, open your Settings, use the search feature and search, sleeping. It should take you to these lists and you can do this on a Samsung as well to reach these lists. I just prefer to share the long route when I know all of the steps to help users become familiar with their devices.
Let us know if this solves your issue and if not we can help you some more.
Best wishes.