At the moment generally with Phoenix Point it doesn’t overly feel as if the Lovecraftian horror aspects of the game has a great deal of impact.
One of the reasons for this is that our forces are quickly and easily progressing to a point where they feel superhuman, so what if through time they also became a little bit more fragile?
You could have damage which causes a drop in the base stats for a solider. For example
Panic during a missions leads to a permanent reduction in willpower
Damage to arms leads to a long term reduction in strength
A damaged leg leads to a reduction in mobility
You could also introduce the possibility of permanent negative perks.
Soldier A has spent too long in the gym, he has a random change to flee from the enemy.
Soldier B was attacked by a face hugger, whilst it was removed and they look okay, there’s a chance over the next x amount of time that they might mutate into a Pandorian.
Soldier C has experienced too much panic, they now have a chance disobey orders on overwatch/reaction fire.
(Those are just by way of examples, I’m sure that balance would need to be considered and that there could be many more.)