littlebit

You can add compost, peat moss or sand to clay soil to improve the texture.  Gypsum will help break up clay soil, also. The drainage is poor in clay soil but on the other hand it holds water during drought. Roses grow well in clay soil. Also, coreopsis, black-eyed Susans, purple cone flower, asters, day lilies, viburnums etc. Just google plants for clay soil. There are a great many that do well. Stay away from plants that require good drainage.  Even if they make it through the summer the roots will probably rot during the winter. 
 
PS Everything in my yard grows in clay soil. Come look!

I think that you would be ill advised to use coal ash on the soil. While there are some nutrients in it, it is contaminated with heavy metals and other things that are bad for plants. I don't suppose it would kill the tree immediately but it might get it eventually. Coal ash has a high pH and dogwoods like an acid soil. 
Wood ash is just fine and I use it myself. 
I can't think of anything gardenwise that it would be good for.