Re: Unplayed podcast downloads keep being deleted
You may want to check that your iPhone has sufficient memory / storage space free in which to store the downloaded podcasts: ie: check in Settings > General > Storage. Eg: An iPhone needs at least 8Gb free in which to operate smoothly. Preferably more, especially if you perform intensive tasks (like video-editing etc.).
NB: If you need to create more space, it is always wise to back-up your iPhone data first. If needed: The linked article below has tips on managing your storage space:
If you need more space for an update on your iPhone, iPad or iPod touch - Apple Support