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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:

https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

It is one of several species of Armillaria. This will be a terminal illness to the shrubs. Trees and shrubs that have contracted this fungus will die, relatively, quickly.

Here is an article for more information:

https://www.gardeningknowhow.com/plant-problems/disease/armillaria-root-rot-control.htm

That would be one of the many species of Clavaria. This fungus is busy decomposing lawn litter and grass clippings. This should be considered beneficial to the lawn.

The only way to get rid of them without destroying your lawn will be to take away their food. This means de-thatching your lawn a few times per year, or otherwise removing any debris as it falls onto the yard.

It is a real pain to control them, and no good reason to do so. They replace all of that yard maintenance, while feeding your lawn once the material is decomposed.

Here is an article for more information on keeping mushrooms out of your lawn:

https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

All debris, like dead or rotting wood, leaves and sawdust should be cleaned away from the area.
Here's an article with more tips: https://www.gardeningknowhow.com/ornamental/fungus-lichen/removing-stinkhorn-fungi.htm

Do you have an apple tree and a cedar? It looks like the fungi from cedar apple rust. If it's isn't that, it is a type of fungus growing in grass. Here are some possibilities:

https://www.gardeningknowhow.com/ornamental/fungus-lichen/removing-stinkhorn-fungi.htm