In this instance, a mire is a place within a marsh, or "cienega," in which the overlying grasses and tules are not strong enough to hold the weight of a person or animal. Although it appears to be as strong as the rest of the grass covered soil around it, if an animal tries to walk across that part, it will break through and become stuck in the underlying water and mud. It usually required two or three men on horses to pull a cow from a mire.
The Spanish word for this type of mire is "atascadero." The Spanish word for miry is "atascoso." (D.G.)