Try using pelletized GYPSUM to break up the clay. It's a good source of
calcium, too, which is often overlooked as a nutrient for healthy plants.
The beauty of gypsum is that it won't raise or lower the pH of the soil; use
as much as you need. . .which may be a lot, from what you say. You should be
able to find pelletized gypsum in 50-lb bags for about $5 or $6 at a garden
center (I don't think the big-box stores carry it). Meantime, don't give up
with adding grass clippings, manure, etc., since you can't go wrong with
such additives in the long run.