Is this a kind of training software that actually exists? This may seem like a silly thing to want to do, but I have been thinking lately that some kind of training software that does the following might be helpful to me:


Show me a board position, and let me study it for a bit.
I click a button or something, and the position is no longer displayed.
After some (hopefully configurable) period of time, it displays the next position in the move sequence.
I have to identify the move that was made that got from the first position to the second.


The reason this is something I'm interested in doing is in the interest of time management during tournaments.  If I want to get up and leave the board during a game (to use the restroom, get something to drink, just stretch my legs and walk around a bit - whatever), I never do it on my opponent's time - because if I return to the board and they've made a move, I can never tell what move they actually made without trying to peek at their scoresheet, or asking (which seems rude...).  So I end up only ever getting up and leaving the board when I'm on move, with my own clock running, and this seems sub-optimal.  This is something I've never been good at (being able to identify what move has been made if I don't actually see it happen), and I'd like to be better at it.

Is there anything available that's intended for building this skill?
I'm not going to answer your question directly, but will attempt to address the root cause of your problem. The skill of seeing which move your opponent played in your absence is something that you should acquire naturally over time. In the beginning I also struggled with this, but after a few months I found that I can easily pick up which move my opponent played. This is probably due to:


Becoming familiar with popular moves in your opening choices.
Anticipating opponent responses. It goes without saying that you should have an expectation of how your opponent will respond whenever you make a move, which cuts down on the number of potential moves you need to actively remember.
An improved ability to keep track of all the pieces on the board - in the beginning, 32 pieces feel like a lot, but over time your mind start to keep track of the individual pieces.
Improvement over time at spotting common tactical themes/motifs/patterns. Doing chess puzzles will probably help in this department.


In normal playing conditions you should never ask your opponent for anything, but you have a right to look at his scoresheet at any time, so don't feel uncomfortable when taking a peek if you are unsure about which move he played. When it is your move (as it would be in this situation) it is OK to even take his scoresheet physically and study it. I see this happening all the time.

A quick note on your habit of leaving the table when it's your move - I believe it is now against FIDE rules to leave the board while it is your turn - you may want to make sure about whether that rule applies to your local competition to avoid any unpleasantness.