Basically, Mockingbird took a much harder stance on superheroing and fell back on her SHIELD/spy training. Hawkeye felt like he had to "save" her, bringing her back in line with the less violent standard superhero posture--"Avengers don't kill" and all that. But Mockingbird resisted because she has issues with feeling powerless, and she even started seeking out bad influences like Dominic Fortune. Ultimately, Hawkeye realized Mockingbird just didn't want to be saved; she didn't really want to go back to how things were or, he surmised, to be with him anymore. So he left the WCA (World Counterterrorism Agency, Mockingbird's personal task force) and they finally broke up for good.
Also, Hawkeye's name is Clint, not Hank.