As someone who plays a fair number of games, and ends up explaining the rules half the time, I'm occasionally faced with the conundrum of what to do when I forget a rule or explain a rule the wrong way. I am human, and as a human, I make mistakes. Games have a lot of rules, and a lot can go wrong. Here's my 5 step program towards righting the ship after a botched rule.
Step 1: Getting a rule wrong is not the end of the world.
If players are having a good time, let it go. As long as the game doesn't take too long to play, this probably won't hurt anything. Plus, you might discover an interesting variant.
Step 2: Don't take advantage.
If you find that you could take advantage of a rule omission/mistake, don't do it. Point it out, and then fix it in the following round of scoring, or the next game.
Step 3: Reach consensus.
Most players are forgiving by nature. Still, they have thoughts on how games should go. Try to find a solution that benefits everyone. Maybe this means starting over, which isn't a bad thing if you find the rule early on. Maybe this means sucking it up, losing the game, but playing the right way next time.
Step 4: Try to laugh it off.
Don't come across as someone who doesn't care that they made a mistake, but try to be self deprecating. It's the old stand-up comedian trick, if a joke bombs, make a joke about how bad the last joke was. If you're having fun, that's the most important part of the game.
Step 5: Learn from the error.
Take this as an opportunity to learn from a mistake and do better the next time. As a game explainer, you're going to get things wrong from time to time. Do your best to improve your teaching skills, keep players engaged in the learning process, and don't worry about the last mistake. The mark of a great teacher isn't by how few errors they make, it's by the way they deal with the errors they make.
Here's my rule of thumb for what to do based on when I find an error:
In the first 10-20 minutes (depending on game length) try to start over.
After that, insert the rule at the start of a round.
In the last 10-20 minutes of a game, just play the way you've been playing the whole game.
This does vary based on the timing of a rule. If it's something that some players were able to do but not others, then it's best to just play the "wrong" way for the entire game.
The crucial thing that I've learned the first time I play any game is that we're going to do something wrong. First games are learning games. Sure, I'm playing to win, but if rules were forgotten/explained wrong, I won't get worked up over that.