Own your mistake before someone else finds it. Same mistake, totally different story.

The error stays exactly the same size, but who reports it changes what it means. Bringing it yourself makes you somebody with standards who caught a problem early. Waiting until it gets discovered makes you somebody who either missed it or hid it, and nobody can tell which, so they assume the worse version. The delay is what does the damage.

When you notice something wrong, say it the same day, plainly, with whatever fix you can offer. No build-up, no defense. People remember the speed of the admission long after they forget the mistake.