If you’ve made a mistake with a client it’s best to fall on your sword as quickly as possible and take responsibility. It’s important for an unhappy customer to feel both heard and validated. It’s also the first step toward finding common ground regarding potential remedies. Any focus on deflecting blame will take a toll on your credibility and reputation. When you know you’re at fault, simply say, “Yes, we made a mistake, and we apologize. Our performance in this case was clearly not in keeping with our high standards for servicing our customers.”
The next sentence you say should be, “What can we do for you to make it better?" This immediately returns power to the customer because the problem has been recognized; it’s time to find a solution; and, you’re putting the ball back in their court. You may be surprised to find that in most cases customers will not be overreaching with their requests. They’ll usually ask for something that's reasonable.
The other piece of the puzzle is circling back once you’ve satisfied their request to make sure the customer was happy with the remedy. If not, you might need to take additional steps to mend bridges. As long as you’re empathetic, patient and sincere in your efforts to remedy the situation, you may find that your relationship is actually stronger after the mishap than it was before.