These are handled as unmarked miscellaneous quests. The title "Ghost Hunt" will not appear in game anywhere.
After the player has first entered the Hub, releasing the Spirits, the player will activate the Priests of Arkay Ghost Hunt repeatable quests.
This can be triggered in one of two ways:
The player can speak to one of the Priests of Arkay added by this Creation and offer to help deal with Spirits which have started haunting nearby locations.
The Priests of Arkay can be found inside the Halls of the Dead locations within the cities that have them. This includes: Falkreath, Markarth, Riften, Solitude, Windhelm, Whiterun.
When the player accepts their first Ghost Hunt, the Priest of Arkay they speak to will give the player an Amulet of Arkay.
The player can accept up to 3 Ghost Hunts at a time, either from the same Priest or from different Priests of Arkay throughout the Holds.
A Haunted location consists of a Boss Chest which has been possessed by the main Spirit, and up to 10 other random Spirits (although there can be less).
Once the player has banished the Spirit possessing a Boss Chest, any other Spirits that possessed items in the location are also banished, each of those spirits will provide the player with more Ectoplasm - so ideally are hunted down before tackling the final.
After the player has completed 3 Ghost Hunts, the next Priest of Arkay they speak to will reward the player with a random Enchanted Item (either a weapon or a piece of armor).
This loop remains active for the remainder of the game.
Priests of Arkay are present prior to the player engaging with Spirit content - however they will not offer Ghost Hunt quests. Instead they will offer the player healing if they are diseased.