Certainly, your characters - whether they are superheroes are not - should have foibles. They should have problems; they should have things that their powers can't solve. That's what makes them nuanced, interesting characters. They can have intense motivations. They should have intense motivations to do what they do.
In Marvel Comics, the worst thing was always that your loved ones could be attacked, or you could be horribly beaten in a knock-down, drag-out fight, but in the Superman comics, you would be run out of town with people throwing rotten vegetables at you and waving a sign that said, 'Superman, Who Needs You?'