I saw your recommendation of using a post hole digger and filling with compost. the year before last I did something similar. I put a 2" auger bit on my drill and put 2 foot deep holes about every two feet across my entire lawn and then spread compost over the whole yard which also filled the holes. It must have been pretty good because the moles moved in last summer and really did a number on my yard. Hoping the milky spore got rid of the grubs because I don't want them back again. I just spread a lot of alfalfa pellets on the lawn. 400lb on 3K sqft. A little over kill but am hoping it prevents the weeds from taking over the bare spots. I usually seed in the fall but will put some down after the pellets dissolved to prevent weeds. Decided to go chemical free but corn gluten is too expensive. Hopefully the alfalfa does some good. At least it will add a lot of OM which will help.
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Getting ready for the Better World Book kickstarter - February 2019