Could this be your man? The county of his birth looks like Henderson, difficult to read on-line.

T. M. Leach, age 33, born in Henderson[?] County, Tennessee, 6 ft. 1/2 in., Private, Company G, 10th Texas Cavalry, enlisted April 20, 1862 at Cornith, Mississippi, discharged for disability June 23, 1862

M323: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Texas