I don't think so. I don't think you can customize anything on the My Work page except for what is in the tiles (done through layout templates).
However, in the Mobile App, it saves the last "Work" area you were on. Personally, I use that more since in my role, I'm hardly ever in the My Work area. I guess it just depends on how much you lok at your phone (for me, I'm always on it. LOL!)