Jumping about 5 steps ahead, exactly how desert is Australia? What I'm getting at is, could we have a "Khanate of the Southern Continent?" where Mongols continue to ride horses and live in yurts until discovered by Europeans...
There's a myriad of reasons why this is completely and utterly implausible.
The main reasons are that;
1) horses - and particularly the Mongolian horses, which were adapted to the relatively cold dry climate of Mongolia - wouldn't thrive in the hot dry deserts and hot humid rainforests of Australia.
In OTL, the Mongols already had quite a bit of trouble adapting to places southern Syria and southern Persia, and Australia is much, MUCH worse than, say, Kerman or the environs of Damascus.
And in fact, the Australian climate isn't very suitable for the whole traditional Mongol lifestyle - even the Australian savannahs are simply too dry for the horses, sheep and cattle that the Mongols depended on.
Even today, cattle raising is only possible in inland Australia because of the many Artesian wells that have been made since the late 19th century.
2) there are no economically interesting empires, kingdoms, petty states, city states, villages or settlements anywhere in Australia, so any attempts to conquer or colonize Australia would only cost a lot of money while the Mongols wouldn't gain *anything* of value.
3) even if the Yuan would discover and colonize Australia, then the settlers would be Chinese, along with perhaps some Koreans, Malays and Javanese. There's very little chance that any substantial numbers of non-sinified ethnic Mongols would come along, especially since, as I mentioned before, the Australian climate isn't exactly suitable for the traditional Mongolian lifestyle.