Wanted to make sure I got something right here, since this can be attached to an elite and non-elite enemies or enemies that cannot be defeated, Does this mean in those cases you are just out of luck and are doomed to take a horror damage for the rest of the game?
The first things that come to mind is Suspicious Orderly or the Hunting Horror. Is there anyone more common enemies that would be included? If so that'd really be rough since it seems the only way you can discard it is by killing the particular enemy. If you get it with the concealed cards in Scarlet Keys that's also really rough.
Edit: Another one comes to mind is Harbinger of Valusia, since when its set asside it keeps its attached damage, and possibly attached cards? In which case you'd keep the horror damage indefinitely.