Best match for your 1880 J. M. White looks to be 1870 Kaufman TX. Looks like the same man in 1860 Jefferson AR -- except for the odd New York birthplace. If these are correct, then it looks like Jackson had 2 or 3 children with first wife Cyntha and Nancy was a second wife.
Need to do some more digging to try to confirm these records.

1870 census, Texas, Kaufman, page 52a
White, Jackson 40 M W AL
Nancy 28 F W AR
Robert B. 13 M W AR
James M. 11 M W AR
Mary L. 7 F W AR (seen with J. M. and Nancy in 1880)

1860 census, Arkansas, Jefferson, Whiteville, page 837
White, J. M. 32 M W NY
Cynthia A. 22 22 F W IL
Robt. B. 4 M W AR
James M. 1 M W AR
Barnes, Jessee D. 27 M W IL (perhaps a brother of Cynthia?)

Jackson M. White bought 40 acres of land in Jefferson AR in 1857.

1850 census, Arkansas, Saline
Burns, John M. 43 M W VA
Levina 43 F W KY
Mary E. 20 F W IL
Jesse D. 18 M W IL
Henrietta E. 16 F W IL
Cynthia A. 13 F W IL (possible wife of Jackson White)
James A. 11 M W AR
Cerilda E. 6 F W AR
Phebe E. 3 F W AR