Certified GKH Gardening Expert
Unfortunately, your photo did not come through. I am unable to identify the mushroom in question.
Before questioning whether to get rid of mushrooms or not, it is important to know their function. Large white mushrooms in open lawn are usually rotters of organic and dead material. Hardly ever do they pose a risk to plants or soil. In fact, they are quite beneficial for adding nutrients back to soil.
I cannot comment on toxicity without knowing what mushroom it is.
Getting rid of mushrooms in your lawn will be easiest to accomplish by taking away what they need to survive. This will be overly moist, damp soil, and humidity.
If you cannot control these factors, then it will be a constant battle. Anything you do to the soil, chemically, will likely cause more problems than they solve.
In short, it is best to leave them, since they are not going to cause harm to your lawn. This article will offer more information: