When and why to set things right once you've made a preaching mistake

Preachers make mistakes in every sermon. Usually, they will be mistakes in delivery. Hopefully, on only the rarest of occasions, they make a factual error.

Sometimes these will slip harmlessly by because they're obscure or inconsequential.

At other times these goofs will be fairly obvious or pointed out due to someone in the congregation being more knowable about a topic than the preacher.

My most public error was at bible college when I used the name of the wrong King during a preaching assignment. 

So, how should a situation like this be handled?

Should a correction be made, and if so, how should it be done? 

Need it involve an apology?

Does the medium of the correction matter?

If there's a correction, what message does this send to the congregation?

Ideally, the mistake isn't a major point of theology - like you've said that there's four in the trinity or you're saved by good deeds - so no ones salvation is in jeopardy of their image of God clouded.

In these extreme cases the correction must be made, publicly, ASAP. If possible you should communicate on the next level up from how you first communicated. If the error wasn't in person, then you should seek to make personal contact, increasingly directly depending on the first medium.

If the point is minor, then I think it can be addressed as widely as the platform you've used. If your mistake was heard purely in the ears of those present, then a simple email blast can suffice. If the transgression went out online, then social media might be the best avenue for clarification.

All that needs to be communicated is what you said, the facts, and a quick apology. No harm, no foul.

But, should you then address it the next time you speak/write?

I can't see why not. If it's just a two sentence correction then it won't take up too much time, but can also communicate a powerful message.

You're not perfect.

You make mistakes.
You're open to correction.

This is especially powerful if you're able to openly acknowledge that a member of the congregation was the one who pointed you in the right direction.

An admittance of your error and apology, even quickly, is both deeply humbling for the speaker and receiver. These words show that the leader is just as infallible, correctable and prepared to be vulnerable before everyone as they humble themselves, with grace offered by those who, themselves, aren't above reproach,