Dear [Name],
I am writing to express my sincerest apologies for the careless mistakes I made. I understand that my errors caused you a great deal of inconvenience and I am deeply sorry for that.
I take full responsibility for my mistakes and I understand that I should have been more careful and attentive. I am aware that I should have double-checked my work to make sure everything was correct before submitting it to you. I understand that this is not an excuse, but I want to assure you that I have learned from my mistakes and I will be more careful in the future.
I understand that my mistakes have caused you a great deal of frustration and I am sorry for that. I want to assure you that I am taking steps to ensure that this does not happen again. I have reviewed my work processes and I am making changes to ensure that I am more thorough and accurate in my work. I am also taking additional training to ensure that I am up to date on the latest industry standards and best practices.
I understand that my mistakes have caused you a great deal of stress and I am sorry for that. I want to assure you that I am doing everything I can to make sure that this does not happen again. I am taking the necessary steps to ensure that I am more careful and accurate in my work.
Once again, I apologize for my careless mistakes and I want to assure you that I am taking steps to ensure that this does not happen again. I understand that my mistakes have caused you a great deal of inconvenience and I am deeply sorry for that.
Sincerely,
[Your Name]