The written version of a mistake is less forgiving than the spoken one, there's no tone of voice to soften it, and it can be forwarded, reread, and quoted back to you. That's a reason to write it more carefully, not to avoid writing it.
Most people either write this email too late, after trying quietly to fix things first, or too defensively, front-loading the explanation before the actual news. Both instincts come from the same discomfort, and both make the email worse. The version that works leads with what happened, follows with the fix already in motion, and keeps the explanation short and honest rather than justifying.
The email that says "here's what happened and here's the fix" reads as someone in control. The one that opens with three paragraphs of context reads as someone still figuring out how to say it.
"Hi all, I wanted to give some context on what's been going on this week, it's been pretty chaotic with a few things overlapping, and I think that's part of why this happened, though I want to be clear it wasn't really anyone's fault specifically, but the client's invoice went out with the wrong amount on it..."
"The invoice sent to the client this morning had the wrong amount, $4,200 instead of $2,400. I've already sent a corrected version with an apology, and confirmed with them that no payment has gone through yet on the incorrect one. Flagging in case it comes up on your end."
The second version is shorter, more useful, and, despite skipping the justification, reads as considerably more credible, because it demonstrates the writer is already handling it rather than still processing what happened. Save the "why" for a real conversation if someone asks; the email's job is to inform and reassure, not to explain in full.
One apology is enough, stated plainly, without repeating it in every paragraph. And close with what's next, even if "next" is just confirming the fix worked. A mistake email with no clear ending leaves the reader unsure whether it's actually resolved or still unfolding, which keeps them worried longer than the situation warrants.
If the mistake is serious enough that a reader might reasonably ask "does this affect me," answer that question directly and early, rather than making them infer it. Most of the anxiety in reading a mistake email comes from not knowing the blast radius, telling people plainly what is and isn't affected does more to reassure them than any amount of apology.