OK. Your test on whether to use the essay should be on whether your essay is mostly about the mistake or mostly about what you have learnt and how this experience has made you act differently in a leadership role recently. As long as it is the latter, you should be fine. The mistake itself should not take more than a few sentences of the essay.
jallen - thanks. Same goes to your example. First glance, Im not sure how your example fits in with the rest of your app. I can see the angle of being conscious about who you align with, but not sure that is a strong example.
And here is my disclaimer

- I was denied without interview last year. So, this advice is second hand - based on feedback from my essay reviewer (HBS alum), and from what I have seen in the 65 successful essays book.
Maybe forcefeed (HBS admit) can add some insight here?