Different kinds of mushrooms can grow in lawns, not just fairy ring fungi. Regardless of whether the mushrooms you are seeing are 'fairy ring' mushrooms or just random, willy-nilly mushrooms, there really isn't much you can do about it. They generally don't harm the lawn (although certain types of fairy ring mushrooms can deplete nitrogen from grass on the inside of the ring and they can make the environment hydrophobic, so water doesn't penetrate to the grass roots).
For mushrooms in general (whether fairy ring or other), If you are concerned about them being toxic to children or pets, it may be a good idea to rake up the mushrooms whenever they appear. They'll keep returning, however, as long as our wet weather continues. Mushrooms recur because the 'bodies' of the mushrooms, which are hidden under the grass, will keep producing the mushrooms (think of the mushrooms as kind of like flowers), and even though we remove the mushrooms ('flowers'), the 'bodies' are still around growing on debris hidden beneath the soil surface. Once the energy source is depleted, the mushrooms will subside. It may take years, however, for the energy source to subside. Just as an example, we removed an old tree from our yard 5 years ago; we still get mushroom flushes from fungi growing on the decaying roots. Over time, these flushes will diminish.
If your mushrooms are truly of the fairy ring type, the grass inside the ring might appear yellow, or the 'ring' might appear darker green. Applications of nitrogen fertilizer will disguise the uneven colors. To remedy the hydrophobic barrier formed by the fungus and get moisture down to the grass roots, you can try using a root feeder attachment on garden hose, and per Todd Weinmann, Cass County Horticulture agent, "punch holes a foot apart in the yellowing or dying area and pump lots of water into the ground to a depth of 10-24 inches. This is necessary because the mushroom growth causes the soil to become impervious to water. Watering with a sprinkler does very little good for the grass in fairy ring infected areas." This would only be necessary only if a hydrophobic barrier has truly formed (they don't always form).
i hope this helps. Please feel free to contact me at the NDSU Plant Diagnostic Lab if you have further questions.