There’s an interesting example of this on the Guardian’s politics blog from a while back now. (Scroll down to 09.16). They were reporting on an interview on the Today show with Tory MP Huw Merriman. Apparently, Merriman said:
“I’m deeply troubled by what’s going on. We all know that if the prime minister doesn’t ship up, then they have to shape out and that’s exactly what happened when this prime minister took over.”
The Guardian played this nicely by introducing the quote like this:
Talking about Huw Merriman, the Conservative chair of the Commons transport committee said that Johnson had to shape up or go. Mangling the familiar phrase (it was early in the morning), he said: “I’m deeply troubled by what’s going on. We all know that if the prime minister doesn’t ship up, then they have to shape out [sic], and that’s exactly what happened when this prime minister took over.”
If a print journalist had been doing a private interview without a live audience, they might have decided just to tweak the quote so it said, “We all know that if the prime minister doesn’t shape up, then they have to ship out.”
There's nothing dishonest about this because it was a slip of the tongue and there was no doubt what he meant to say. However, we acknowledge that some people would argue that what a politician says is a matter for public record and so shouldn't be changed.
Another option could be to avoid it altogether. They could introduce the quote as the Guardian initially did…
Talking about Huw Merriman, the Conservative chair of the Commons transport committee said that Johnson had to shape up or go.
....And then, rather than repeating the mangled quote, just move on and quote what he said next, neatly avoiding the problem.
If interviewing a case study, we would have no reservations whatsoever in recommending you tweak the quote to what they meant to say.
A [Sic] refresher:
[Sic] is used to indicate that something has been intentionally left as it was originally spoken or written even though it is incorrect. It’s usually italicised and should always be surrounded by brackets of some description and it signals that a quote appears without edits. It directly follows the error.