Mimosa trees that require corrective pruning can recover and put out new growth. You will need to remove any damaged limbs back to the primary main limbs. Don't leave stubs. It looks like the center of the tree lost the primary limb. Cut it back to an outward growing lateral branch if available. You may want to trim back the other branches a little too just to even up the shape of the tree and to take some weight off of the outward reaching limbs.
I don't know if Mimosa wood burns well. I would guess that it is more of a soft wood than a hard wood like oak or maple.