That hand is Charlie's mom off to the side. You can see that the hand has the same clothes as the mother does. At the end of the song she comes out through the corner which she came from. This slipup is fake!
I was especially interested in this slip-up. So I took my old video and fast-forwarded it to this song. If you look closely, it is not a prop man's hand on the cane. If you watch, it is the mom's because she is wearing some sort of suit like jacket.
Actually, the person that handed it to him was Charlies mother. Look at the sleeve, and then look at his mother in another scene. It's the same color and style as the person that handed it to him.