In the CW Research Database I found one Cutrer from Mississippi:

N S. Cutrer
Enlisted as a Private
"E" Co. MS 3rd Battn Infantry

Additionally, the following men with the Cutrer surname served in Louisiana
units:

Edward W.
H.
Irvin E
Isaac N.
J. P.
L. L.
O. B.
Owen B.
W. R.

Good Hunting,

George Martin