Asking to reattempt research after having a hard time initially Last summer I asked a professor if I can try my hand at doing research under him in his field of interest. As I was an undergrad student (I still am) taking taking summer courses and working (I wasn't getting paid for the research) at the time and lacked a lot of skills/creativity to help in with what he was doing, I feel like I was not very successful in being very helpful with putting together a proper research report. 

This being said, nearly whole year has passed since then, with no plans this summer for courses, I have an idea I want to do original research that could be helpful to his field (I didn't find anything written on the matter, so I'm not 100% sure if it's a good one, but I sincerely think it should be tried if it hasn't to verify its validity for his field) and I in turn gain from his guidance, and possibly get a paid position with him. 

Is it possible for me to get back to doing research with him after having a rough time last year? If so, how should I ask for it? Should I simply write an email asking if its possible for me to try doing research with him again or compose a formal research proposal and give email it to him to demonstrate interest in the field? The risk in turn of that would be that I would waste my time writing a proposal that would never be read or  rejected where my efforts should be elsewhere.
This is somewhat field-dependent, but a lot of undergraduate research is not really for the professor's benefit. If they needed help, they'd enlist graduate students or postdocs. Working with undergrads is more about giving the student a learning experience.

You don't say anything about feedback from your professor, but in any case, I see nothing wrong with asking to work under him again. As for your idea, it's definitely a good idea to run it by your professor before investing much more effort, as he is in a better position to judge whether it's feasible, whether it's been done before, etc.