Dear {Name},
I know it was a small mistake, but I still want to apologize for {mistake}. I pride myself on {different behavior or action}, and I can assure you that no one was more surprised than I when I discovered my error.
Please let me know if there is anything I can do to help mitigate the fallout. I resolve to avoid such errors in the future, and have {action steps to avoid future mistakes}.
Sincerely,
{Sender}
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Index of letter of apology templates