The herbs in the pictures look happy and flourishing.
That's not much info re the "white stuff." The flies buzzing around make me wonder if it is whiteflies that flutter away when you water. They can look just like powder or flecks. But the whiteflies, like aphids and a few other things, secrete a sweet sticky substance called honeydew, that attracts the flies.
Tell us more about what the white stuff looks like. Can you rub it off? is it powdery? How much of it is there?