To 'hand something over' to someone means to pass it to them or to transfer responsibility, ownership, control etc to another person As it implies 'hand me' implies that the item in question is to be handed from the giver to you, or that the item being handed has distance from the person asking. The correct phrases would be; 1) Hand that to me 2)Hand that over to me Both have the same meaning. Hope that helps

To hand something over is to transfer ownership or responsibility for it. You might hand over responsibility for a project to another person. Handing something to someone might also transfer responsibility (eg handing a parcel to the postman) but is often more casual. 'I took a biscuit and then handed the plate back to my mother'. Does that help at all?