The objective of an adjustment letter is to inform the reader that an unsatisfactory situation is being corrected. You should avoid excuses or long explanations about what went wrong. Stick to the facts of the adjustment and your reader will comprehend the corrective action that you are offering or have already implemented.
In the first paragraph of sample adjustment letter #1 the writer apologizes for an incorrect shipment and clearly admits that a mistake has been made. He then provides the details of the solution. In the end he does not reiterate his apology. Once said, leave well enough alone and focus on positive actions.