The end of the endgame that was no endgame

A couple of weeks ago I wrote about an "endgame" in one of my correspondence chess games at ChessWorld. From the comments to that post, I learned that my position at that time should have been filed under middlegame and not endgame. The game just finished with a loss for me: I gave my opponent the chance to play a mate-in-three combination. This did not stop me from winning the tournament, though. The image on the left shows the position before my final mistake. White to move, what is the correct way to avoid mate? See the annotated game for the move I played and the mate in three.

I wrote in the previous post on "endgame" study that I should start some kind of basic endgame study. I actually started reading "Pandolfini's Endgame Course" by Bruce Pandolfini. Right now, I am working through the chapter on minor pieces vs. a lone king. The problems such as mates in four are not very complicated. In fact, I would probably resign if I would have only the king left, so I don't expect to encounter these situations very often. However, I try to solve the problems without a chess board which brings me to the limits of my current chess visualization abilities: everything beyond the next move still feels troublesome. But by not moving pieces during analysis I hope not only to improve my chess endgame skills but also my board vision. And soon I will move on to things like rook and bishop vs. rook endgames, a situation that is actually similar to another correspondence chess game I am playing right now, which gets me back to the question of just-in-time engame study...