Another enemy weakness, which are usually pretty tame. Looking at the two elements, the effect and the discard condition, we get:
The effect: An enemy that needs killing, which is, ironically, just what Tony likes to do. It even comes with it's own bounty, so Tony can engage it for free. It's 4 combat, which is stronger than most enemy signature weaknesses, but Tony should be prepared to dish out 3 damage to such an enemy, and if he's using Tony's .38 Long Colt, it boosts him and rewards him for the kill. It would be a signature asset if it didn't come in with a Doom on it, which makes it a weaker Ancient Evils that you can cancel if you are fast enough or if it gets drawn during the Witching Hour. It has the usual "loophole" that any investigator can deal with it, although the Quarry's Aloof keyword makes that less appealing than some of the "on the board" weaknesses. If you are on a small map, killing this maybe a turn and change. In a larger map, you might just want to take the Doom hit or deploy cards like Nimble (although Tony will need some boosts before that's going to happen) or Elusive to get close, if not in, the space.
The discard condition: Kill it, if you don't feel like ignoring it.
Taken together, this is a below average signature weakness, which is almost more a benefit, except for the slim possibility of Dooming out a scenario.