You did not indicate if this is in a lawn or wooded setting but since you mention a lake house, I'm assuming it might be a wooded area. Unfortunately getting rid of them is not very easy. The mushrooms are the fruiting bodies of fungi. Removing them doesn't kill the underground mycelia from which they are growing. Picking them can improve the appearance of an area, and somewhat prevent their spores from spreading, but most spores are windblown, so they can easily come from mushrooms in surrounding spaces. If they are growing in a lawn area, they could be developing from buried scraps of lumber, dead tree roots, or other organic matter. Their appearance often indicates over-irrigation or poor drainage. Removing excess thatch and aerating the soil improves water penetration. Either way, the only way to help the problem is to physically remove the mushrooms as they appear and/or keep the puppy away from the areas where the mushrooms are growing.
