You do not use TuneIn app to play TuneIn stations on HomePod (or Echo/Alexa for that matter) except, maybe via Airplay. But I don't do that.
I keep a TuneIn account (free) just to know/keep a record of my stations (I have a lot) and I use it in the car too.
But to first play stations on mini via TuneIn or iHeart Radio (might be USA only), just say “Hey Siri, Play station name”. This is then added to the recently played list.
Apple does not let us make our own TuneIn lists. Frustrating. But you can build that list up by voice until, you can then go to the Home app and long press a mini or Apple Music under radio and see your stations you asked Siri to play. You can then just tap a station to play.
Note that the station name may be different. One local station near me needs the city name added after its letters or I get the classical HD channel it runs. So try different iterations if the first is rejected. Looking at the station web site and what they call the streaming channel they run from the web site helps too.
BTW: What are the top Polish Jazz radio stations?
Do widzenia I powodzenia!
