Mushrooms aren't a bad thing. They thrive in moist, well drained, loamy soils which is also the perfect environment for growing a lawn. If you are overwatering a touch, or have been having more rain than usual, mushrooms often appear. Once the water is back to your normal levels the mushrooms vanish on their own.
Mushrooms spores are minute and are blown in from great distances. You cannot control that. These spores can also 'hiberanate' for centuries. Can't do much about that either.
If you really want to get rid of the mushrooms, have you thought of a sign that says -
Please don't pick these mushrooms !!