From this background "IF" Carr's CSR record is correct as being wounded and captured in December 1864 and I have no doubts that it isn't, it would have had to have been in some small action here in Arkansas, which would have been probably somewhere near the Benton-Pine Bluff-Little Rock arena.
It could have well been that the 2nd Arkansas Cav. was one of those units who were detached from Prices main body during his retreat and were attempting to get around the Yankee's in Central Arkansas when he was wounded. We don't know when in December that Thomas Carr was wounded and captured. But in any case whatever happened to Thomas Carr a record survived probably from Union POW records. He may well have been seperated from his Unit and maybe at home when a Yankee patrol came by.
These are all possibilities.