A)The post mentioned the "200 poor children in county." The 1840 Census luckily breaks down by race, sex and age. To give you and idea of what percentage of the county's children were considered poor. Looking at school age children, for New Kent it gives a total of 359 white males aged 5 through 14, and 307 females the same age.(The school system of course was only available to whites.)
B)Some background on the public school system, such as it was, of the time.
Charity or Public Schools- . . . The lack of funds, as we have seen, was the cause of the failure of Jefferson's [education] plan of 1796, and this law[school law of 1810] said that all money coming into the state treasury from fines, forfeitures and certain other sources should be set aside to provide schools for the poor children in every county. The money thus set aside was called the "Literary Fund." In 1816 the money loaned by Virginia to the United States government in 1812 to help carry on the war with Great Britain was repaid to Virginia, and the General Assembly added this money, amounting to over $1,200,000, to the Literary Fund. Beginning in 1818 $45,000 each year was paid out of the interest on this fund for schools. Later on the amount increased as more fines came in.
Only the children of poor white people could get the benefit of this money. In 1825 for instance, 10,226 children went to these schools; in 1851 31,486 were sent, and in 1859 54,232 were sent, the money coming annually from the fund for the schools being about $160,000. The schools were charity schools and wrongly called public schools. They were open only about three months in the year and nothing but reading writing and arithmetic were taught. Especially in the eastern section of Virginia it was considered a disgrace to be so poor as to have to go to the "public schools" and long after they had ceased to be charity schools and had become schools for all classes, rich and poor alike, and good enough for the richest as well as the poorest boy and girl, the "public school" was looked down upon in some parts of Virginia because the old idea of charity school still stuck.
-School History of Virginia- Edgar Sydenstricker, Ammen Lewis Burger-1914