I didn't find any nasty white gunk under the black scabs, either. The "kernel" attached to the scab was all I encountered, and that varied in size, from a centimeter to far smaller. I tried to remove all scabby material from around the crater after lifting off the black scab, leaving a glistening crater of red, moist tissue that certainly didn't appear to contain any pus/infection that the body creates out of white blood cells to repel bacteria.
I wish it had been this simple for us. Had there been a kernel attached to the scab I would feel much better. Since it didn't have one, I am second guessing everything we are doing (or not doing)!
Love BYC - at least I do not have to muddle through it all alone and I have cute emoticons to express my chicken exasperation!