No, I couldn't unpublish it and then re-publish it, I have to leave it the way it was when I entered it into the competition. I can't say exactly why that's the rule, but it makes sense when you think about it. The competition page keeps the links to all the games that were entered, and if you changed the game after the competition was finished, then people who read the game after you've edited it won't actually be reading the game you entered into the contest.
For instance, somebody could enter a ten page troll game and get last place. They could then edit the game they wrote and make a really good game. Then one day, somebody might read through all the entries in the competition and think, "That game that got last place is really good. Why did it get last place? The competition must be unfair."
That's not a particularly good example, but in the case of my game, the voters gave me an average score of 6.5 out of 10. I'm editing the game to make it a lot longer and more detailed, and I have a lot more time to proof-read and stuff. Hopefully when I'm finished, the game will be worth a lot more than a 6.5 out of 10. People thinking of entering the competition might read the updated story and think, "Why did that story only get a 6.5 out of 10? The competition must be really, really hard. I don't think I'll enter." ... So there's a good example of why I'm not allowed to edit the game I entered into the competition.