Contact the lost cause again with a plan of how you have—or will—address their specific concerns. Don't forget the most important part: Start talking to them about upcoming jobs you may be able to help them land.
Concerned you've made a mistake so grave that there's no way the customer will take you back? Consider this story from "How to Win Friends and Influence People." There was once a pilot who took off and immediately experienced engine trouble. He barely made it back to safety. Apparently, a ground crew member had put the wrong fuel in the tank. The pilot found that technician and said to him, "I want you to work on my plane exclusively from now on." The crewman was shocked, but the pilot explained that he wanted someone who was meticulously careful. He knew that, moving forward, the crewmen would be just that.