I was quite impressed with the narrative told in Jack Weatherford's 2004 book, Ghengis Khan and the Making of the Modern World, and I thought it was an interesting and enjoyable read, but the wikipedia page on it warns it has strong revisionist components.
The claims of the Mongols heavy role in bringing printing, gunpowder and the compass to Europe were, in particular, surprising to me.
But I had always had the conception of the Mongolian military as cavalry based, yet Weatherford emphasizes the might of Mongolian siege weapons and how that military siege capacity allowed Mongols to take fortified cities, which cavalry alone would not be able to penetrate.
So what were Mongolian innovations in siege weapon design and use? Did Mongolian siege weapons ever use gunpowder like a primitive cannon in some way?
Ok Mongol seigecraft is not my best area but here's what I've got. The Mongols undoubtedly used gunpowder on occasion. We're not sure how widespread their use was. We certainly know that they were used but we're not sure how often and where they were used. Were they used only occasionally or were they an integral part of the war machine? In East Asia we do see evidence of widespread use, albeit mainly in the crude bomb form, hollow iron and ceramic balls filled with gunpowder and launched from trebuchets. But evidence becomes a lot more dubious once we look at other areas such as the Middle East. There the best we have is references in texts to weapons that might be gunpowder based but equally might be simple incendiary weapons.
In terms of other innovations the only thing I can think of is the introduction of giant counterweight trebuchets to Chinese seigecraft in roughly 1268.
If you an assessment of Weatherford's other claims/a run down of what did the Mongols actually do I direct you to my post here on Mongol government, my post here where I discuss Mongol death tolls and did anything good come from the Mongols. If you want more reading check out the book list linked on the side of this sub.
The Mongolian invasions that started under Chinghis Khan began in Asia (roughly)thusly: First the conquering of the other northern steppe peoples, then sweeping down into western China, then the conquering of the Jin in the north of China (who themselves were semi-steppe peoples/non-Han Chinese), and then the Song in the south (the actual Han ethnicity).
In the steppes, cavalry warfare was favored and pretty much solely used because their enemies were similarly cavalry based. Once the Mongols moved down to Tibet and western China, they faced the beginnings of fortifications and suffered more losses than in the northern steppes. The Jin used their city fortifications and their own horse archers to fight against the Mongols for a time, before the government retreated south, forsaking Beijing and leading to their demise. The Jin are recorded to have used siege weapons against the Mongol invasions
Their enemies in the south, the Song, were the ones who had extensive use of gunpowder. It was after the capture of Song and Jin peoples that the Mongols then had a wider access to siege weaponry.
After the Mongolians had broken up into a sort of Khanate confederacy, Hulegu sought to decisively put the Middle East under Mongolian control. Accounts of sieges on Muslim cities talk about the use of non-counterweighted "catapults" and counterweighted "trebuchets", ballista type devices, as well as rocket-aided weaponry. This is discussed here in the "Hulegu Moves West" section.
According to Timothy May's The Mongol Art of War, the engineers used to build/operate the machines were not Mongols, but rather Chinese (Jin) and Persian. These engineers were either hired or conscripted by Mongol armies to siege cities.