It probably has to do with the relationship between education and poverty. It is not that they don't have access, but that they don't have the education to understand what foods to eat. You can live pretty cheaply without resorting to fast foods or processed foods, because those are actually MORE expensive than good foods. You can get a bag of rice and vegetables very cheaply.
There are also a lot of poor skinny people. If you are very fat and poor, it is probably a decision-making problem or lack of knowledge. They either don't realize the processed foods are more expensive, worse for you or they lack the cooking skills to prepare the other foods.
It's just not cost effective to feed your family on processed foods or fast foods, given a low budget.