1. I doubt you can shoo them away "permanently" - they seem to simply be taking a step (or two sometimes) back, so if they are already at the "just spawned" position they end up off-screen, but they come back anyway because they go back to advancing on Karryn afterwards.
2. They do have some HP, maybe even the same they would have during normal battles: when my Karryn kicks them for 300-400 HP, they take damage and step back (same as shooing), but remain, yet when she punts them with a 3k+ crit - away they go.
Personally, the point of shooing the goblins away is lost on me: why would I spend a turn on the goblin if I risk gaining
nothing (worst case - they remain where they are), whereas I can let them come closer and punt them to make those handshake sessions "goblin-free"?
P.s. What I hate the most about the whole thing is that both actions become unavailable whenever Karryn is Horny. And I have no control over the state because it keeps popping up randomly ever since the "Open Public Toilet" passive which just blank cheques it with "Chance of getting Horny every turn increases". I did NOT expect an effect like that to work anywhere outside the side job required to get it, much less a different side job.
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nature of it is...well - the image is damn accurate.