No.
Don't just point to the dietary composition and think that alone explains it. If calories in > calories out you get fat; it's that simple, and it happens no matter what you eat.
Can a "better" diet make it easier to be in balance, sure. Can a better diet make for less non-obesity related health issues? Definitively. But there's no magic bullets.
On a low budget, you have to be rational about what you chose, especially when you're on a budget to feed others. IE; avoid unnecessary luxury food, and look at how many mouths you can feed with the budget offered. (I count meat as a luxury btw... it isn't cost-efficient if you look at it rationally, and over-consumption of cheap meat comes with health issues of it's own.)
I can't be arsed to do the numbers, but look at how many calories a day that one month supply represents. If it is reasonable, it's a cost-efficient way to ensure that a single human being has enough food to get through a month. If it's too high, it's possible to cut down on the amount and feed more with less. If it's unreasonably low, you definitively can't say it's a reason for them getting fat.
The poor are fatter than the rest because statistically they have less education, and know-how about what they should be eating. It doesn't help to provide them with a perfect diet and expect them to magically be in good health in all ways if they lack the knowledge required to stay healthy.
The only real way to help the poor is a combination of good public education and social mobility. It doesn't help those who are poor right now, but it changes things in the long haul. The poor right now are fat because they can afford to eat more food than they need. There might be more or less legitimate reasons why they end up eating more than they need, but you can't violate the laws of thermodynamics that it all boils down to: if calories in > calories out you get fat