The Civil War itself impacted the children of slave owners in that many did not have to go to war, or see their fathers go off to war for the Confederacy. The so-called "20 Negro Law" allowed exemptions for one white mail from conscription for every 20 slaves owned. Thus, larger plantation owners could exempt themselves and potentially all of their male progeny from service.

In terms of Reconstruction, there isn't specific scholarship that I am aware of that is solely dedicated to the impact on the children of former slave owners, but we can deduce the impact from the historical record. This impact would have varied by socioeconomic status of the slave owner, and by location.

Many southern plantation owners had much of their wealth tied up in their slaves and in their land. This wealth passed from generation to generation. Slaves and land were commonly willed from generation to generation. Thus, an immediate impact is that many of these children of former slave owners simply saw large parts of their family's wealth, and thus their inheritance, vanish in to thin air as their emancipated slaves exercised their freedoms.

It also meant that the children of their slaves weren't, as were their parents, slave owners presiding over large plantations, but in essence landlords in what eventually became sharecropping situations or large-scale employers of many freedmen. This meant, for them, essentially, less profit for the same amount of work, because they were now paying freedmen or sharecropping with freedmen, rather than pocketing all revenue from their cash crops or resale of food crops.

The 20 Negro Law also created, essentially, two separate class of white males following the Civil War: those who served and those who did not. Many who served were from lower income households, and many who did not were from large, slave-owning plantations. It simply served to exacerbate the existing gap between rich and poor. That gap helped shape politics in the South to some degree, as the eventual agrarian populist movements that took root in some places like Texas were descendants of the poorer whites who did not own slaves or owned less slaves.

While the children of former slave owners were indeed impacted socially and economically, your question reflects a larger one, in that the difference between the children of slave owners and poorer whites in the south went on to define politics in the South for the better part of a century.

Sources: Wilkison, Kyle G., Yeomen, Sharecroppers, and Socialists: Plain Folk Protest in Texas, 1870-1914, Texas A&M University Press; Sitton, Thad and Utley, Dan K., From Can See to Cant: Texas Cotton Farmers on the Southern Prairies; Cullen, David O' Donald (ed.) and Wilkison, Kyle G. (ed)., The Texas Left: The Radical Roots of Lone Star Liberalism;