The best MK song has a great melody: Done with Bonaparte!
He seems better with the folky melodies, all the way back to Local Hero.
Mark himself said in the early 90s he never wrote any good melodies, and of course, it's not entirely just being humble, as he compares his melodies with amazing songs like "Blue Eyes Crying in the Rain", and it's what he probably meant — singing melodies, as in not instrumental melodies. Mark doesn't have classic melodic standards like this, some writers are just more melody-driven than others, especially if you have a great singing voice.
It's probably a bit easier to write good melodies to sing if you're Paul McCartney with one of the best, versatile, wide-ranged and record-friendly voices in rock's history than if you're a mumbler like MK or Bob Dylan. Because of this, certainly with Mark's music, the last thing you think about is the melody, I can't even remember any melodies from the top of my head, to me it generally feels like spoken word anyway. It's not a bug, it's a feature.