Dear {Name}
I'd like to apologize to you and everyone else who was part of the video meeting on {date} at {time}. I was struggling to maintain a professional workspace while dealing with {domestic matters} at the same time, and I forgot that {my mic wasn't muted/my camera was on}. That said, I never should have {said/done inappropriate thing} in the first place.
I assure you that my conduct will be entirely professional from here on out. I am very sorry to have behaved so indecorously and hope you can forgive me.
Sincerely,
{Sender Name}
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