Interesting situation. I don't know much about the fungus side of things, but I'm sure someone else will be along soon to talk about that.
It does remind me of a situation I had once, where everything rotted and got the white fungus on it. Instead of dealing with the fungus directly, I decided to go a step further and try to find out why the fungus liked that location so much. I discovered that the drainage wasn't very good. This may or may not be your issues, but what I did was to take the top soil, move it to one side, fluff up the subsoil with a fork, then build raised beds. Put down a layer of
compost (in my case raw kitchen scraps) then replaced the topsoil (which had been laying in the sun for a few days). Never had the problem again. Like I said, your situation may be different.
Getting rid of the offending fungus is one option. Have you given any thought as to what you will replace it with? It's a bit like pasteurising
milk - if you just leave pasteurised milk in the fridge it goes off, but if you put some yoghurt in it, it last longer and goes a delightfully sour taste instead of putrid rot. I'm thinking about the soil, one thing I've seen is people kill the offending fungus/bacteria only to have it come back. Other people take steps to replace it with new bacteria and this seems to make a longer lasting solution. Some people even skip the step of killing off the offending stuff, and just add new fungus/bacteria/microbes/whatever, which drive away the unwanted pathogens. I like this last method because I'm lazy and it skips a step.