To avoid a geographical POD which is kind of a messy thing, how about we say the Mongols never retreat? It was unnecessary and a poor decision considering they were winning, but they were afraid the Japanese were coming back with reinforces (that the Japanese didn't have). Let's say the Mongols advance further into Japan and burn the capital and execute both the Shogun and the Emperor. Of course, the Mongols are going to have more fighting once they reach the mountains inland, where a lot of Daimyo will be hiding out, ready to rebel. Some Daimyo will surrender and be kept as vassals by the Mongols. Perhaps they even allow one to be the de jure Shogun.
Japan will likely be annexed to the Yuan Dynasty, anyway. Sadly for Japanophilies, Japan is an insignificant island nation in medieval times. Heck, not even a really united one. If Korea gets annexed, you can sure bet the Japanese do too.