Every mistake made is a good lesson learnt. Only foolish men let the same mistake repeat itself time after time. Win your clients over with your sincerity; they will appreciate you for your good intentions and the potential of becoming even better in future.
Look at it from this perspective. When you make a mistake, you tell yourself you learn from it and never let it happen again. Your clients can also do the same because the moment you screw up, it also become their mistake........ by giving you the job. If they heed your advise to learn from their mistake and never to let it happen again, do you think you still can work for them in future? Its really tough out there.