Mesquites, and other undesirable (e.g., invasive) can be a major challenge.  Click here for a good summary of Mesquite challenges.  Effective treatment can require a combinaton of mechanical (e.g., grubbing) and herbicide.  Click here for a Texas Agrilife "Bruch Buster" overview of herbicide treatment.
Click here (overview) and here (leaf spray) and here (stem spray) for information on Mesquite removal
Summary: Click here for info.  "Early-day ranchers like W.T. Waggoner called mesquite “the devil with roots” because it absorbs all of the water in its surroundings, causing other plants and trees to wither away and die. This, in turn, allows more mesquites to move in and take over. Last but not least, this tree has positively vicious thorns."