MPlayer disables powersave for your screen when it has a video to play, which is what you want usually, but not always. In particular, the fact that powersave is disabled regardless of whether the video is running or being paused can be somewhat irritating if you’re watching something in multiple takes and just keep it paused in background for a long time to avoid writing down position at which you stopped watching.
The problem
There isn’t really an issue with MPlayer here, because it has a relevant option, the problem is that multiple attempts to find a solution online have failed due to a phrasing issue. Also, it seems that not a lot of people try to achieve this.
The solution
The solution was eventually found in the place you should consult first, which
is relevant documentation (man mplayer in this case). However, same story as
online, you won’t find what you’re looking by searching for “power” keyword.
So it was essentially a guess that led to the solution in this case.
The option you need is called -nostop-xscreensaver.
One might think that it controls screensaver only, but apparently it does more than that. I guess, MPlayer (and other players) never control display power explicitly, only through screensaver interface, so looking for an option with “power” in it was doomed right from the start.