While awake, can thought activity be transferred between the subconscious and conscious awareness, by will?
I don't think the scientific literature would necessarily describe the process as "shifting thought between subconscious and conscious awareness" but putting exact definitions aside, obviously this does occur.
When I recall information, in some sense information is being moved from the unconscious to the conscious part of my mind. By focusing on a different task I am able to switch my attention to something else. Thus, what was previously the focus of my consciousness no longer is.
You could think about top-down versus bottom up processes of allocating attention, where top-down refers to where attention is driven more by the individual and bottom-up refers to when attention is driven more by environmental cues.