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

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.