I don’t think that Libspotify, Spotify’s unloved and unmaintained library that mopidy-spotify uses, supports podcasts as they are a relatively recent addition to the platform.
If you can find a podcast uri through other means (e.g. from an official client or maybe from mopidy-iris or mopidy-mopify?) then you could try and play it directly in mopidy-spotify, I have no idea if that works.
We would like to add support for features like this. We can access the search etc through Spotify’s other Web API and if it turns out libspotify can play them then someone could implement that.