Answer: The mushrooms are a natural part of the decay process and will continue to grow until conditions become unfavorable for them. Eventually they should exhaust themselves, or may disappear when the weather turns drier and warmer.
You might try (experiment on a test area first) topping off the mulch with another type of material (other than wood based) such as shredded hardwood bark or cocoa shells or even straw and see if that is enough of a change to stop them. Alternatively, you could remove and replace the mulch with a different kind, but again, experiment on a test area first. You might also consider a fence to keep the dogs away from the mulch. Last but not least, you might find an expert who could identify the mushrooms and tell you if they would be poisonous, or not. In the case of mushrooms, there is always a possibility that a small child might eat them, too.
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