Own the mistake before anyone finds it. A confession is a story you tell. Getting caught is a story told about you.

Speaking first changes what the mistake becomes. Told by you, it comes with context, with what you are doing about it, and with the fact that you came forward. Discovered by someone else, all of that disappears and the headline becomes the hiding rather than the error. Most people forgive the first version far more readily than you expect.

If something has gone wrong and nobody knows yet, use the hours you have left. Go with three things: what happened, what it affects, what you are doing next. The conversation is short and the relief afterwards is out of proportion to it.