Thing is, better food is available to the poor as well, you just have to be willing to put in the work for it. Buy vegetables and make salads instead of spending the same amount of money at McDonald's, for example. The price of fresh vegetables at Walmart has never been out of reach even for someone working 40 hours for minimum wage. Housing might be ridiculously expensive, and medical care if you don't have insurance? Good luck. But basic vegetables? Rice and beans? (Which make for a complete set of amino acids, BTW: there's a reason rice and beans is such a popular dish in Central America). Those have stayed affordable even when the price of other things has gone up.
Now, I'll grant that there are plenty of poor people who are drinking soda and eating junk food. Not going to deny that. But I have always been able to go to Walmart and buy lettuce and tomatoes for my salads, and I've never seen the price of those basics skyrocket like the price of eggs (at one point) or meat have. So the poor people who are drinking soda instead of water, and eating chips instead of salads? They're choosing those foods, not being forced into them by poverty.
There are plenty of areas where rich people have a big advantage over poor people in terms of access to things that provide longevity. But food, at least in America (the only country whose food prices I'm familiar enough with to talk intelligently about), just isn't one of them.
Now, you could argue that poor people didn't grow up with parents who taught them how to cook healthy food on a tight budget. Yes, that's true for many (not all) of the poor (again, at least in America, I don't know enough about other countries here). But there, it's not being poor that's keeping them from eating healthy, it's not being taught. Money isn't the limiting factor there.