You won't be able to eradicate the mushrooms, since the mushroom itself is just the fruiting body of the fungus that produced it (think of it like a flower or seed head on a perennial plant), and the rest of the "body" of the fungus is hidden underground and weaving through the soil or possibly the wood of the tree's roots. In that analogy, removing flowers from a plant won't kill or remove the plant itself, and the plant might produce more if it's still ready to flower; this is why removing mushrooms won't affect the fungus itself other than stifling its ability to reproduce.

Some fungi that appear in this situation are harmless to plants, in that they are only decaying already-dead matter in the ground (or the mulch itself), recycling its nutrients so plant roots can absorb them. Mushrooms will stop reappearing when the fungus' food source has been depleted, or when weather conditions change (like the soil dries out more, as most fungi thrive with moisture). Since we're in a prolonged drought right now, the only areas we'd expect to see much mushroom growth is where irrigation has been used or where the soil isn't drying out much (heavy clay, perhaps), but scattered rains might be enough to allow some mushroom development.

If the mushrooms are arising from a fungus invading the tree (and it might not be easy to tell if they are), then the tree should be evaluated by a certified arborist or licensed tree expert. They won't be able to cure a fungal infection or halt wood decay, but they can help narrow-down a diagnosis and may be able to determine if enough internal wood has degraded to make the tree fall risk.

If you are concerned about a child or pet trying to eat any of the mushrooms, since there isn't an easy way to tell which are toxic to ingest, then you can certainly pluck or scrape them off with a gloved hand or shovel and dispose of them, but they may reappear for a time until the fungus is done reproducing.

Miri