어쩌다가 and 어떻게 하다가 are about the same. 어쩌다가 is a shortening of 어찌하다가 (어찌하다 + -다가), and 어찌 of 어떻게, so the two are essentially the same. The difference between 어쩌다가/어떻게 하다가 and 어떻게 is the presence of (하)다가. which conveys the sense of something unintended happening in the course of some other action. e.g. 망치질을 하다가 손을 다쳤다 = I hurt my hand doing hammering work. Because of this accidental nature, 어쩌다가/어떻게 하다가 usually describes something that has already happened. 어떻게 on the other hand doesn't have such a restriction so it can be used for anything, including the future tense and how one should go about something. The meaning of the 어떻게 part is the same between the two. It can mean "how?" or "in what way?" in a question, or "somehow" or "in some way" in a statement. Examples: - 어쩌다가/어떻게 하다가 돈을 잃어버렸어요? = How did you end up losing the money? - 어떻게 하다가 이렇게 됐는지 나도 모르겠다 = I myself am not sure how it came to be this way. - 어떻게 하다가 보면 잘될지도 모른다 = By trying different things, we might it work out (rare present/future example). - 앞으로 어떻게 할지 의논해 보자 = Let's talk about what we are going to do.