I don't think it's about a break-up at all. To me, it reads as being about two broken, lonely people trying to shake off their pasts and be together. The compare scars, hurts, and broken hearts. He is honest with himself about his feelings for the first time ("took years, my baby, just to come clean") and they agree to try a relationship. ("Come see about me whenever you're lonely")
I don't think it's about a break-up at all. To me, it reads as being about two broken, lonely people trying to shake off their pasts and be together. The compare scars, hurts, and broken hearts. He is honest with himself about his feelings for the first time ("took years, my baby, just to come clean") and they agree to try a relationship. ("Come see about me whenever you're lonely")