The whole table decides who gets to trigger their haunt, so its something the GM have to prepare in advance (or ideally, be surprised by their own too using improv). The player that is haunted has the spotlight and will be based on the haunting questions they answered in character creation. So yes its possible the mission gets harder and they may just choose to retreat if it happens, or be able to beat them, eitherway, its pivotal moment for their character in the campaign, its okay to lose or avoid it if the players decide to, but that ultimately means they will still be haunted. It will only be when they faced it and resolve it will it full be gone and unlock their signature skill.