Can I redraw my hand at the start of the game in Sentinels of the Multiverse?
No, the rules do not allow for mulligans and the official site has no provisions in any expansion or clarifications that add this feature.
The official sites downloads page has copies of the official rules and the rules for each expansion. None mention a process for mulligans of your initial 4 card hands. The Changes section also only notes changes to the Villian, Heroes, and Environments cards and no changes to the turn structure.
From the rule book (page 4)

Game Setup
[...] After the heroes, villain, and environment have been chosen, each player draws the top four cards of their hero deck. The game begins with the villains turn, then proceeds clockwise around the table [...]