I am sorry that I neglected to {make the copies/mail the package/file the files/etc.} that you requested.
I know that this clerical task was my responsibility. One factor was {mitigating factor or explanation}, but at the end of the day, I should have delegated or otherwise made sure it was completed.
Please let me know if there is anything I can do to help rectify my error. I'd be happy to {reach out to client/go to post office/stay late/etc.}.
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Index of letter of apology templates