TL;DR: No, this discrepancy is never explicitly dealt with, and in fact, most of the later material appears to operate in a universe where that specific element of the first movie simply never happened.
I have not seen Highlander: The Source but I have seen the other 4 movies, much of the TV series, and the first few issues of the comic book. I also found an animated series (a lot of it is on YouTube and it's not bad) that is set in the future, but includes Connor's backstory. As far as I have seen, the media is split into two distinct timelines:
- The first, second, and third movies where Connor defeating the Kurgan ends The Game and wins him The Prize.
- Everything else -- the fourth and fifth movies, the TV series, the animated series, and the graphic novels -- where the second and third movies did not happen, and The Game simply did not end in the first movie. No mention of Connor winning The Prize appears in any of these media.
For the first movie (and the nameless fiasco), there was no Duncan yet, so those two movies operate on the assumption that Connor really was the last Immortal on Earth and really did win the Prize. The third movie was made after the TV show began airing, but it too largely ignores the issue; instead, The Final Dimension reveals that some living Immortals "hid" from the Gathering, and only when they reemerged does Connor's immortality kick back in and the Game resumes with Connor again winning the Prize in 2024.
In the pilot of the TV show, which supposedly happens in 1992, we meet Duncan McLeod, another immortal that Connor has known for centuries. As Duncan was first killed in the 1600s and was clearly alive in 1985, there is no way Connor was the last living Immortal on Earth. When Connor appears and explains the Game to Tessa, including the One Rule and the Prize, he expresses horror at the idea of certain Immortals winning the Prize. Connor never reappears on the TV show and nothing is ever mentioned about it again. It's clear from the beginning that Connor, as far as the TV show goes, did not win the Prize in the first film, but that otherwise the show is supposed to be a continuation of the first movie (and ignores the second and third).
Similarly, the animated series incorporates the first movie into its own continuity as past history, but it makes no mention of Connor winning the Game. Instead, the Immortals all mutually agree to stop playing the Game to deal with a major catastrophe, but the Game is still on (and one Immortal is still trying to win.)
The comic book series (the first one at least) is also intended to follow directly from the events of the first movie, and they also ignore the fact that the Kurgan's defeat should have ended the Game. Instead, Connor merely received some sort of "evil Quickening" that was different from a normal Quickening but not the Prize.
The fourth and fifth movies follow essentially from the timeline of the TV series, with both Duncan and Connor appearing as their respective characters and The Game still well underway. No mention is made of Connor ever winning the Prize, and in fact in the fourth movie we find out that The Watchers are actively trying to prevent anyone from winning, by secluding them at the Sanctuary.
(I haven't read any of the second comic series but AFAIK it is supposed to be entirely a prequel to Highlander and, as such, wouldn't shed much light on the subject.)
Overall, it seems that the sanest and least stressful way to interpret Highlander canon is to:
- Pretend that Connor won something other than the Prize in Highlander
- Ignore Highlander II and Highlander: Final Dimension
- Pretend that all subsequent media follows immediately on the heels of a retconned Highlander