What do I do when I make a mistake? Own it. I first let the right people know that a mistake was made and what it affected. I also let them know I'm working on a fix or that I need some help making the fix.
I don't normally make mistakes, though... It's not that I'm some sort of genius that doesn't make mistakes. Nah... nothing quite so arrogant. My secret is that I follow the same rules that I laid out for everyone else. I submit to a design review not only by the users involved but by peers, the SME on whatever I'm working on, etc.. I submit to a code/peer review just like everyone else. I develop and test in dev and have someone else deploy my reviewed code to QA (we have the rule that no one promotes the code they wrote and the auditors absolutely love it). The same goes for staging and production. We all do the same thing and if it's a production "urgency", then the same thing happens but at a greatly accelerated rate (sometimes in as little as 10 minutes depending on the problem). If the fix can only be tested in prod, we do so on a hardware snapshot of the database that we have ready to rock at the drop of a hat... any hat.
The 2nd to the bottom line is 1) no matter who caused the problem, I'll jointly own it with the person that made the mistake or own it entirely if I'm the one who made the mistake and then 2) time proven procedures are expedited by 3) people that are rehearsed at 4) following the procedures/rules and 5) doing their job.
The bottom line is that because we all work as a team (me, the devs, QA, the window/infrastructure folks, the users, and the managers), it's a very rare thing for any of us to deploy a problem to production. The auditors like the fact that we can prove all of that with documentation we enter in the ticketing system, source control, and the code itself.