First, if the mulch was just added, it may not be the source of the mushrooms. It could contribute by raising soil moisture, allowing the fungus the conditions it needs to fruit. Depending on the kind of mulch, though, and whether it was decomposed or decomposing, it may have been brought in with the mulch. Mushrooms will use available resources and then disappear in most cases. If it seems unsightly, chances are it will not be there long or only during periods of the moisture and temperature being just right. Mushrooms rarely impare plant growth and never attack the living plants in a planting, generally.
Fungi, mushrooms especially, are hard to eliminate without harsh chemicals. Changing soil pH can stop their growth, but it helps to know a number of factors first...what pH the particular species prefers, what it's feeding on, etc, but as I said, they're most often a short term issue, often fruiting only once or twice a year and in specific seasons and conditions.
If it is just more than you can tolerate, you could try removing all of the mulch and trying a different mulch, wood mulches can be a food source for mushrooms, but not all woods, and the freshness or decomposed state of wood mulch is a big factor in whether it provides a ready source of critical nutrients or not.
Many mushrooms are more likely to grow on decaying stumps, roots, or woody matter already present in the soil, any fruiting bodies of mushrooms on fresh or partially decomposed wood mulches often fruit lightly at first and then more heavily and they're done, or from heavily to more sparsely until nutrients are used up, at any rate, aside from major underground sources of wood decaying, like extensive root systems and stumps, the current mushrooms species may never recurr, and at worst, may recurr seasonally once or twice a year and otherwise never fruit for the remainder of the year.
A sulfur or copper spray may suppress some fungal growths temporarily, but may be a detriment to the environment to an extent exceeding anything the mushrooms could cause and could be no guarantee of any control or more than a very temporary control.
