One Punch Man and Mob Psycho are very similar shows by the very same author, but have you ever wondered what would happen if each one took the other's formal and did a spin on it? That is precisely what these specials were, short skits with self-centered mini-stories that revolves around Saitama and gang, sometimes other characters that never got the amount of screen time they deserve (Puri Puri Prisoner). I can see why they did not have this content cramped into the main story, they do jump off the timeline of the main plot quite often, your typical sitcom that doesn't respect the nitpicks ofthe plot, but it's fine for what it is.