clay bulked with organic should still be pretty "sticky" and stay in place. But by adding organic material, you would help water penetration and air in the soil for plant rooting. The trick is that netting, but if it's still pliable you should be able to aerate and topdress without ripping it up too much (depending on how much of it is exposed). worst case scenario, you use a step down aerator or hydraulic plugger (instead of a wheeled) to keep from pulling the netting or very worst case, a pitchfork. If you can open it, you can get an organic topdress started.
That's the slow solution. The quick(and expensive) solution is to rip it up and do it right from the beginning.
When you reseed, be sure to choose the most shade and fungus resistant species of grass that you can find in your area.