Fungus growing in the lawn

This is going to be nigh on impossible to stop with any chemical. It is almost certainly related to the tree roots below that are rotting away. Once it has done it's intended job in nature of breaking down the wood, it will disappear.

The only sure fire way to stop it is to excavate the rotting wood from the roots (a massive job that probably you wouldn't want to undertake) and re-turf or seed.

It sounds unlikely to be the species of fungi associated with the dead thatch in the lawn that people often call fairy rings. None of these have hard fruiting bodies as you describe your fungus.

I'm afraid I agree with Fractal on this.

poisen or use something to get rid of the rotting remaining parts of the tree

the point about the tree roots is that they exist in the soil, not that they are alive - it is the rotting process which causes the fungal growths, as Fractal says above.