- Can I use The Home Front to 'move' a damage onto Poltergeist, given it says 'cannot be damaged'? No: "To be or to become damaged would include anything that would place, add, or move damage tokens onto the card."
Monster. Geist.
Cannot be damaged except by Spell, Relic, or encounter cards.
Parley. Test (3) to attempt to banish the geist. If you succeed, deal it 1 damage.

Who knew that Fine Clothes would make Exorcisms so much easier?
In the card pool up to and includng Path to Carcosa Deluxe expansion, we've seen only 4 player cards (not counting different XP versions of the same card) that can harm the Poltergeist:
So unless you're packing very specific Mystic cards, your only option is to exorcise these ghosts via Parleying. The good news is that a decent intellect - or, as above, some Fine Clothes - make this process not excessively onerous, although one you'll almost certainly have to do twice.
As they say, dress for the job you want. In this case, that job is apparently Father Merrin.
Given the age of the last review, I thought I'd update the list of Relics and Spells available to damage this. In broad categories and not counting upgrades, they would be:
0 XP, Damage on Any Success:
1+ XP, Damage on Any Success:
- Spirit Athame
- Ornate Bow
- The Hungering Blade
- Timeworn Brand
- Blood Eclipse
- Shards of the Void
- Song of the Dead
1+ XP, Damage on Other Effect:
- Ancient Stone - Requires a previous purchase/identification
1+ XP, Special:
To summarize, Mystics have plenty of options to deal with this, and Guardians aren't completely dead in the water at the start, thanks to Enchanted Blade. You may still want Fine Clothes if you can't reliably fetch your blade, potentially replacing it as you go. Later on, it's more up in the air, but Guardians may want to veer towards Timeworn Brand. While The Hungering Blade will also do the trick, its reliability means it's not for everybody.
Other classes didn't have the same damaging options available, so they'll want to focus more on tools to boost their . Evasion is still an option if you're able to avoid running back into it after you're done, and Waylay could help you bypass the fight altogether.