Here are some pictures for those of you who have some input. The hole was easy to dig, it took about 5-10 minutes. The soil?, dirt? is relatively soft. The grass isn't growing very deep roots though. I think that is the main problem or the effect of the problem. We haven't put anything on our lawn because I'm new to this and would hate to poison my children with some icky stuff.
So, could the cover crop of radishes help? Would it be better to dig holes and add compost as suggested in other topic areas?
Oh and sorry for the typo above, lawn, not law.