I was going to give a similar answer to Kappo. Right now, the United States is considered the "most obese" nation in the world. Like the first answer implied, not consuming fresh, healthy, and seasonal produce and protein leads to poor eating habits and eventual health problems associated with these habits.
But I don't believe it is simply the "inability" to purchase healthy food that causes people in poverty to develop what has become America's epidimic. One of the speculations for the causes of this country's health issues arising from weight gain is our larger than life fast food network and availability. It has been written about in articles and books and even documented in film - interviews with people living at or below the poverty line, who, when given the choice between purchasing $3 worth of produce or a super value meal at McDonald's - are going to go for the super value meal. Why? Well, it seems like more food and fills them up better.
The long term problems of course are much costlier than purchasing the produce in the first place. I think largely, our country - for as advanced as we are in so many things - has failed to connect nutrition to progress. And poverty is one of the first factors connected to a lack of good nutrition.
A couple of resources you might want to look into (if you are interested in this subject even further) are:
Food, Inc (movie)
In Defense of Food: An Eater's Manifesto (book by Michael Pollan)
Unfortunately, poverty is often accompanied by lack of education in some key areas which also affect health and development. While it is money which often keeps people from eating healthier, as mentioned above, it may also be lack of basic nutrition information. It may also be that children's emotional and intellectual development may be stunted by the lack of proper stimulation. For example, parents may not read as often to their children or encourage them to read for themselves. Because poverty is one of those things which helps create low expectations, those may be set too low; and, without challenge, there is little motivation for improvement or excellence. If the parents see no way out of poverty, it's likely they're unable to help their children see a way out. Poverty, research tells us, is a cycle. Once a person is born into poverty, it's too easy simply to stay there--and continue the cycle into the next generation.
One of the ways that poverty can affect health is by changing the way that people living in poverty have to negotiate their habits, etc. If you cannot afford to purchase produce as many many urban families cannot, you increase the chances of your children and the parents as well ending up with diabetes leading to immense health care costs and changes in lifestyle, etc.
Even if the children don't end up with diabetes or other related health problems, it is very difficult to concentrate or do well in school while suffering from incredibly poor nutrition and dealing with the effects of those problems.