The finish is just getting old and dried out. Try rubbing the spots with rubbing compound like you use on a car and then apply polish. Just don't use Turtle wax brand compound. Use something like 3m or Metuiar's. It shouldn't take too much elbow grease doing by hand with a rag.
If you think there is a chance someday you might refinish the secretary stop using aerosol furniture polish especially pledge. They contain silicone which goes through the finish into the wood and stays there. When refinishing the silicone can make the new finish bead up like water on a freshly waxed car. Google fisheye in a finish to see what I mean.
You could use paste wax or car polishes to get away from the silicone.