In FireRed/LeafGreen, there is a girl in Lavender Town who tells you she sees a
'White hand' on your shoulder. Does this have to do with the storyline, or is it just for the effect?
it have no affect to your storyline at all so go on with your yourney.
The same girl is later shown on Skyarrow brigde in BW2, after completing the story line. Apart from that, though, it's only added to make the game more interesting.