Normally the jamb is fitted to be flush with the wall. On the door shown, the jamb was fitted to be flush with the vertical board on left. I don't understand the reason for leaving the gap, but here's how I would fix it. I would set a new board on top over the door to come out flush with the jamb. Next I would apply a new wide molding around the door. The molding will span the gap on the left and hide it completely.
I'm assuming you have another vertical board on the right. Not shown in your pictures. If not, you will have to install another board to get flush with the protruding jamb and molding on top.
If you don't have time to do it right the first time, when will you have time to do it over?