Even if the Mongols could make gunpowder, gunpowder is best utilised in guns, which probably are going to be beyond the Mongols.Who says the Mongols can't take advantage of gunpowder?
I'm not saying this is entirely feasible, but we do see an example in OTL in western-central Asia of the Mughals - they were more than just steppe horse archers.
What? The Mongols used gunpowder weapons. Not from horseback, admittedly, but they totally used artillery and hand-cannons as soon as they could get their hands on them.Even if the Mongols could make gunpowder, gunpowder is best utilised i n guns, which probably are going to be beyond the Mongols.
China remains fragmented long enough after the collapse of the Yuan dynasty for a new set of Mongol dynasts to re-establish control over at least part of China, maybe? Some sort of "Northern Yuan". Simply having a Mongolian dynasty in control of (at least part of) China up until 1800 is cheating, but it should fulfill the demands of the OP.
A Northern Yuan that controls Beijing, then? Or at least something more than Outer Mongolia.This is already OTL. After the formation of the Ming Dynasty a "Northern Yuan" continued to exist until the 17th Century, which claimed to be the legitimate ruler of all of China. After the construction of the Great Wall it only ever controlled the Mongol steppes, but it counts.
That's certainly fulfill my demandsA Northern Yuan that controls Beijing, then? Or at least something more than Outer Mongolia.