Ok, I don't think we have a trope for this: in old videogames (platformers specifically), has anyone noticed that the "hero" is usually really weak? Take the early Mario games as an example, Mario can only attack by jumping (he can't punch or anything), is killed by merely touching an enemy, and until Mario 64, could never take more than two hits without dying.
The original Crash Bandicoot games are a more recent example: he can attack, but never has more than two hit points, and can be injured just by walking into things.
I was thinking maybe something like Paper Hero
or Wimpy Hero