I believe that if you comes from a poorer or improvised background you will work as hard as you can in school to make sure that you can rise above your situation, I believe that many students realize that a key to a successful life is by getting a good education.
When I was going to university, I met a lot of students from India who came from very malnourished and impoverished areas of the country. They didn't care about anything other than success, because being here in the USA they knew what hard work could result in. So yes, they do, and because they have reason to
An impoverished student may or may not work harder at their studies, I think it really depends on their frame of mind and how their impoverishment has effected them. You can theorize that they may work harder if they've come to find that they may be able to better their situation with a proper education. In this way they may work harder because of the promise of a better life in the future. Of course, it could also go the other way. It's possible that the impoverished student may be too concerned about other things in their life to concentrate on their studies. In this case they may be too hungry or work while not in school, which can negatively impact their studies.
This would be impossible to generalize across groups, and is more likely to be influenced by other individual factors. Things like support systems can go a long way to getting a child motivated to better themselves, while things like lacking basic necessities or dealing with major issues coudl completely deflate a student.
No, there is no research suggesting that all poor students work harder. While many poor students do work very hard, there are also affluent students and middle-class students who work extremely hard. To research this, it would be necessary to specify terms as far as "impoverished" and working "harder" at studies.