It is more likely to be down to the input from the parents, or in the case of poverty lack of input. The basics are that poor parents are on average less well educated and don't have either the inclination or knowledge to give their children quality input especially in early years.
The is also a growing problem with the children of wealthy/educated parents who are too busy with their jobs to give their children the quality input they need to thrive.
Diet has very little or nothing to do with it. Any sensible early years teacher could tell you this in an instance. If you arrive at school knowing your letters, being able to count to 10, recognizing your name when written down, being able to hold a pencil properly etc. you have a massive head start in life and this can NEVER EVER be closed by anything the state can do in the educational system.
The reality is that the children of wealthier and/or better educated parents are more likely to turn up on their first day at school being able to do all those things. The cost of getting your child to be able to do these things is minimal and "poverty" in the western world is not a barrier to achieving it either.
Even when we get to school the attitude that the parents hold to the value of an education and behaviour of their children has a huge impact on the how well a child will perform throughout their school career.
Poverty is a symptom of low educational achievement on average and it breads low educational achievement on average. How you break the circle is difficult to know, but throwing money on diet and/or the educational system won't work and does not work.