This is my first attempt at a treasure hunt. I'm in an agency, paid the money to get the information, but the Priest has no interest in letting me do a favor for him.
First you open the hunt (+10 with the correct faction, pay the price for the info)
You will be told who the townie is that was last known to have the object.
Then you do your cases like normal. You have to hope that the person who has the object needs a favor when you have one coming to you. If not you will have to "burn through" that favor and 'rebuild' (do the 3 cases) to earn another favor. Hoping that the new favor will be for the townie that you want it to be for.
Hunting involves the agency working to solve as many favors as possible. Becuase it is random which favor you will get next those that do the most cases tend to do better in hunting.