Dear [First Name],
 
I am writing to apologize for [specific action or mistake that caused the need for the apology]. I understand that my actions have caused inconvenience and frustration on your end, and I am deeply sorry for any harm that I may have caused.
 
[Elaborate on the situation, acknowledge the impact of your actions, and express regret for any harm caused]
 
I want to assure you that I am taking steps to ensure that this type of situation does not arise again in the future. I have taken [specific action that you have taken to prevent the issue from happening again].
 
[Offer to make things right if possible, and reiterate your commitment to providing excellent service]
 
Please accept my sincerest apologies for any inconvenience or frustration that I may have caused. I value our business relationship and hope that we can continue to work together in the future.
 
Thank you for your understanding and for giving me the opportunity to address this issue.
 
Sincerely,
 
[Your Name]