What is it and is it okay to leave in our yard or keep removing?

We live in the mountains of Arizona. During our monsoon season, mid July through mid September, we get most of our rain. A few weeks into the monsoon, these gross brainy looking thing come up from the ground. What are they? I remove them and throw them away. Should I be leaving them? Could them be good for the soil? We get mushrooms too that I remove, but I though I heard they can be good for the soil.
Coming up from the ground.
Pulled up {with shovel, eak} very small roots.
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