It sounds like you were making payments, but for less than the amount that was due - causing you to be late for two months. If that's the case, it's unfortunate, but those late marks are accurate. The good news is that you caught the situation quickly, and did a really smart thing to prevent it in the future. You also set yourself up for a small interest rate discount if these are federal loans, since you are now using auto-pay.
A goodwill letter is a request to a lender to adjust their reporting due to the circumstances that caused the negative information on your report. In your case, you might explain the mistake, and the steps you've taken to ensure that it won't happen again, and ask them to remove the lates. The lender isn't required to do what you ask, and might not be willing to help, but some will respond positively.