Here is what I find on a quick look see:

William Potts aged 26 in 1860 enlisted in Co K 3rd Ark Cav in Yell Co Arkasnas on 2-28-1862. He was killed in battle on Oct 3 1862 at Corinth MS

Thomas (aged 17 in 1860) enlisted first in Co F STirman's Bn with brothers Joseph and Richard but was left and listed as absent in Ark. He enlists again in Co E Gordon's Ark Cav at Dover on June 27, 1862 and is listed as present on last muster in 1864.

There are 2 James Potts (age 13 in 1860) either one could be the son... James W. enlisted as Pvt in Co H 26th Ark Inf in June 1862 in Johnson Co Ark and was killed at Battle of Ark Post Jan 1863

another James Potts enlisted in Co C 21st Ark Inf at Clarksville, AR on Nov 18, 1861 and fell overboard on Steamer St Francis #3 eight miles below Memphis on APril 13, 1862 it does not say he drowned or survived but just never heard from again. Could have swam and returned to Ark and enlisted in 26th but no proof or anything.

I know he had sons Joseph, Wm, Richard, Thomas, James, and John. There is a JOhn Potts but since Kirkbride's JOhn was only 11 in 1860 probably not him.