Dear {Recipient}
I owe {you, institution, the team, company, etc…} a sincere apology for {state situation}.
It was never my intention to create such an awkward and uncomfortable situation. I hope that I can be of assistance in correcting my mistake and resolving the matter.
While I have caused this unpleasant experience, I am able to see where I went wrong. {State the cause of the mistake if possible}. I believe acquiring this insight will allow me to learn and grow and prevent similar issues in the future.
Please feel free to let me of your thoughts and concerns over this matter. You can reach me at {email, phone number}.
Sincerely,
{Sender}
{Sender}