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For the mesquite wood, all I do is remove any loose bark and then toss it into the pit. That's AFTER it's well seasoned (dry).
Keep in mind, mesquite has a very heavy flavor and will sometimes over power your food with its flavor. I'd start with a known neutral type of wood and add a little mesquite and see how the taste goes. If it's good, but a little lighter flavor than you want, add a bit more mesquite on the next cook.