I was playing as Expatriette in a Sentinels of the Multiverse game and my starting four cards were all ammo cards. As each ammo card says it has to be played next to a gun card, I effectively couldn't do anything with my hand.
Do the official rules allow a hero to trash and redraw their starting hand? If so, is there a penalty? I searched the official rulebook and the Rook City expansion book and didn't see anything, but I don't know if one of the later expansions or official rulebook clarifications added a provision about it.