We can't identify the mushrooms to exact type since there are many that can look nearly identical between poisonous and edible species (and we don't comment on edibility for that reason). Mushroom ID can involve microscopic characteristics and chemical tests that can't be determined from photographs. Still, the simplest solution without needing to have an ID is to use gloves to pluck or dig up those you find to discard them before allowing the dogs into that part of the yard. The "body" of the fungus producing the mushrooms lies belowground in a root-like network of fine strands and can't be dug up, and may keep producing new flushes of mushrooms when the weather is suitable, at least until the fungus' food source is depleted. Many fungi like these are consuming already-dead plant matter in the soil, including dead old tree roots and stumps, and aren't of concern for live plants in the area. Actually, they tend to be beneficial because they're recycling those nutrients back into forms plants can absorb. Mushrooms are comparable to flowers or fruit on plants -- picking them doesn't kill the fungus (or plant, in that comparison) and more can be produced at a later time. No fungicide will be effective, nor is one necessary, and as our weather dries they will stop reappearing for the time being.

Miri