How do you try to save a client relationship after messing up repeatedly?
Client relationships are usually not personal. They see the work you do and want that look on their brand. Treat this as a professional by showing them how they will achieve their goals using you. You can always offer a free test, but most importantly, tell them what matter-of-fact calculated new system will improve the situation. Get yourself to that level, and then with humble confidence, express how you will do it right. In the comments, please share examples of this dynamic and what you did.