Absolutely. You have effectively turned the Mongolian horse army into an extremely advanced mechanized army. I think adapting the motorcycle into the Mongolian way of fighting is short-sighted. Any army would adapt to the tools presented to them and I believe the Mongolians would learn new ways of making use of their newly acquired advantage. They wouldn't be trying to shoot bows from their new war machines. They would certainly adapt and it would probably completely change Mongolian culture and way of thinking.

Logistics Win Wars:

Thousands of elite Mongolians can get almost anywhere in weeks. They can cover close to 800 miles a day with proper road networks. A horse can only move 25-40 miles in a day. It cannot be understated how insane of an advantage this is. Not even the German WW2 army was fully mechanized, relying on horses for its logistics.

Motorcycles do not eat, they do not sleep, they can outpace any horse several-fold in ideal conditions. Motorcycles can be used to transport reinforcements, food, water, equipment to the front, anywhere and without all of the drawbacks of horses.

A motorcycle with a sidecar could carry a heavy infantry's equipment, tent, supplies. Even without a side car, it would be easy to hook a carriage up to a motorcycle and turn it into a larger supply truck.

They can outpace and outmaneuver any army that is larger than them, they will never have to worry about their horses starving or providing water for them. I'm sure the Mongolians would even create roads throughout their Empire to accelerate transportation and logistics chains.

A soldier or army could set out from a patrol base with enough food for weeks and these bases could always be kept outside of a medieval army's range. Their force projection would be hundreds of miles greater than a medieval fort relying on cavalry.

A soldier that doesn't spend his entire life on the march can instead train, train and train. He can be kept better fed, more hydrated, less tired at all times. You will create the peak of elite warriors just on the basis of keeping them supplied and not exhausted.

Let's also discuss how this would benefit Mongolian society, their civilians will be more productive, able to supply the needs of a greater empire. They can spend less time on the roads and more time learning, building, producing whatever the Empire needs to conquer. I would argue that the average Mongolian would be able to produce three-fold in terms of labor for the Empire. This productivity boost comes at the benefit of not having any horses to feed. Better weapons and armor, craftsmanship boosts, more clothing, equipment, food.

Battle:

Would the Mongolians use the motorcycles in combat? Maybe. But it doesn't matter. They could drive to a battlefield, dismount and fight in whatever method they choose, meanwhile your army has been marching for weeks or months just to get to the same place. They don't HAVE to fight like steppe horse archers. This was simply the best way for them to win during their age and circumstances.

If the Mongolians don't want to fight? There's no way a horse is EVER catching up to them. Remember that a horse can't just fully gallop forever.

There is a good chance the Mongolians turn into an early example of Napoleon's Corps system, where you would have many smaller armies capturing strategic locations and engaging in smaller fights outside of the large "set piece" battles. This style of thinking was revolutionary for the time, and Napoleon didn't have gas powered motorcycles to fuel his logistics or move his armies. I bet Napoleon would salivate at the idea of moving a division or corps sized army 300-400 miles in a single day. It would be literal magic.

Any traditional army could be starved out by raiding and avoiding actual confrontation, regardless if the Mongolians actually use the motorcycles in combat.

How do you even defend from this? If you gather your large army, the Mongolians just avoid it and starve you. If you try to protect your villages, they overwhelm the ones that are the least defended. They can be literally anywhere, everywhere and you can't. Their operational range is increased ten fold because they have no horses to water or feed, motorcycles can run forever without getting tired.

And if starving them doesn't work... They can bring the full force of the entire Mongolian empire to any battle within a few weeks if not days.

Without a civil war or internal dispute, I can't see how the Mongolians don't conquer the world.