Put on rubber gloves. In a bucket, combine one part tree herbicide with three parts dish washing soap.
Stir the herbicide and dish soap with a paint stirrer, mixing the two ingredients until well blended but not foamy. The consistency should be like soapy water with few bubbles.
Fill a spray bottle with the herbicide and dish soap mixture. Spray the mesquite tree pellets or sprouts liberally with the mixture until soaked thoroughly.
Allow the mixture to seep into the mesquite tree pellets or sprouts as long as directed by the manufacturer. If the directions call for more than one application, do so as suggested on the herbicide label for best results.
Cover the mesquite tree pellets or sprouts with a tarp while still wet with the detergent and herbicide mixture. This will prevent rain from watering the mesquite sprouts and block out any sunlight.
Leave the mesquite covered with the tarp for a two- to three-week period then remove it to see if the pellets are dead. Dig up the fledgling plants with a shovel to eradicate the tree.