The Mongol-Goryeo forces were pretty close to taking all of Kyushu in the first invasion, but it failed when the Mongol commanders chickened out and ordered the retreat to the boats, eventually being sunk. The second invasion, on the other hand, had almost no chance of succeeding, with the Japanese being ready.
If the Mongols hadn't boarded their ships during the typhoon of the first invasion, they could have taken Kyushu in that attempt. There seemed to have been a clash between the Koreans (who advocated staying on land) and the Mongols (going back onto the ships), and OTL was decided by the Mongols. With the Japanese forces having their back to the wall (only one fortress remained), an intact invasion force would have been able to conquer Kyushu and perhaps head for Western Honshu.
The one change which would be needed to have the first invasion be successful (not necessarily conquering Japan, but succeeding in invading Japan): Don't get on the ships. Really, the thing was that Kyushu was about to fall, with reinforcement being weeks away. One more push by the Mongol-Goryeo forces would have meant complete occupation of Kyushu.
Should the invasion had succeeded, I can see Korean influence being dominating in Japan. The unique position of Goryeo within the Yuan sphere had allowed the Goryeo king to have the title "King of Shenyang," a basis that would later result in Yi Songgye's rebellion when he was sent to invade Liaodong. Given that the most important fighters of the Mongol forces were the Koreans, and it was the Goryeo King at the time (Chungnyeol) who ultimately pushed for the invasion of Japan, I can certainly see the Goryeo court having great influence over the affairs of the conquered Japanese islands.
The question is, how would the Goryeo court have influence in the conquered parts of Japan. I can see three ways to this:
1. Trade/Cultural exchange: With the subjection of Japan, the threats from the Wakou pirates would have decreased, and as such trade and cultural exchange would most likely have been higher. Korea would most likely have recovered far faster from the damages of the Mongolian invasions, and the higher population would only have to face the dangers mostly from the north. Also, influence would naturally grow.
2. Population movement: The "King of Shenyang" title was originally given because of the large amounts of Koreans who moved to and lived in Liaodong. If Japan is occupied, I can see a sizable number of Koreans going to Kyushu, and setting up communities. This means that Koreans would be setting root in Japan as never before since the fall of Baekje, and that in itself would be another strength for Goryeo.
3. Title: It was the Goryeo King that pushed for the invasion, and as such he would have a say in the administration of Japan. Even a small role would have large repercussions, and there is always the possibility that he would have received an honorable title concerning Japan. This would most likely link Korea to Japan. Add in possible intermarriages between the royal families of Yuan, Goryeo, and Japan, and you have in-laws connecting from Dadu to Kyoto.