I'm not particularly interested in some "studies" that come out with statistics saying its hard not to be obese when poor. The fact of the matter is that it's possible to eat relatively healthy affordably (e.g. rice, potatoes, food on sale/discount), and that if you were truly poor you should be saving money by eating less food.
Or your assumption wrong. Here is a reasonable explanation: People are poor since they got bad self control, which also makes them fat. People who lack self control are easily tempted with shitty fast food, so their areas mostly serves it rather than real food creating these "food deserts".
If there was demand for food in those areas people would sell it, but there isn't.
Edit: A strong piece of evidence is that people aren't getting poorer, but they sure are getting fatter.
If we're talking about the obesity rate, statistics make it clear that it's easy to be obese when poor, not that it's not easy to not be obese for the poor.
Or are you talking about statistics that asked if poor people tried to not be overweight but couldn't do it? If so, could you please send link to that - as I'm not aware of any such large-scale study and quick search didn't reveal anything significant?