Both of these characters have caused Superman no end of torment, and a case can be made for either as his most constant cosmic-level enemy. Difference being Mxyz only wants to torment the Man of Steel for pure fun; Darkseid does so basically because he's an embodiment of evil Tyranny, and Superman being an embodiment of Hope is abhorrent to him.
Supes and Mxyz have never really met in a full on "physical" battle; the Imp most often can't even be touched by the hero. Darkseid's "avatar bodies" have faced Superman time and again, sometimes whipping the floor with the Kryptonian with ease... other times getting his over-sized, overbearing, overly confident Apokalyptian @$$ handed to him with gift wrapping.
I'm uncertain if Darkseid and Mxyz themselves , however, have ever truly battled... but considering both are extra-dimensional beings of near limitless, barely fathomable, incredible power (Darkseid enough to exist outside the Crisis reboots, Mxyz enough to destroy the very concepts of words and their meanings from existence)... I wondered who would (most likely) win if they went all out. Both beings have shown capable of manipulating aspects of Reality, Space, Energy, Matter and Time, but to different effects. I do know that when Darkseid found out Mxyz was part of the simplified Anti-Life equation, he went either mad or depressed.
Has there ever been a definitive battle between the two?
If so, who won?
It may be somewhat speculative, but are there any feats which may fairly show their comparative power levels to each other?