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.