Hi cynda
You may already know that there were still Harveys in Stanton St Bernard in 1871 but no mention of White Pits. I had a look at the images of the 1851 census and found households for each of Daniel, James and John Harvey but no mention of White Pits there either. Puzzling!
I did think it might be a reference to Lime Kilns but can't find anything about them either

If you haven't already got copies of the images for 1851 or 1871 and would like them, send me a PM with your email address and I'll forward them to you.
Angela