Question

How does the government contribute towards poverty on access to free basic education?

Answer 1

No one can study with an empty stomach. You need at least fulfil their food needs or every man capable of working will be sent to the field. If the government can erase poverty, the citizen can start to look at other than fulfilling basic life needs like education. 
Enough food is also needed for brain development. Poverty in neonates and child can cause impaired intelligence later in their life.

Answer 2

If a government does not facilitate access to free basic education, then it's poor citizens will not have a chance to raise themselves up out of poverty. This is because free basic education helps children to learn the ABC's and can instill in them a love of learning and knowledge which can give them impetus to find a way to continue with education in which case they will be better prepared to lift themselves out of poverty by learning the skills to become say nurses, teachers, scientists etc.