Dear (recipient's name),
I apologize for the misunderstanding that has apparently taken place. I didn't know you were unhappy with the service in our department. I, unfortunately, reacted out of frustration because of the rumors I had heard around the office. I know I should have come to you and taken care of my problem directly. I ask for your forgiveness. I will strive to be more open in my future communications with you and your department.
Sincerely,
(your name)