  We investigate a strong version of the integral Tate conjecture for 1-cycles
on the product of a curve and a surface over a finite field, under the
assumption that the surface is geometrically $CH_0$-trivial. By this we mean
that over any algebraically closed field extension, the degree map on the
zero-dimensional Chow group of the surface is an isomorphism. This applies to
Enriques surfaces. When the N\'eron-Severi group has no torsion, we recover
earlier results of A. Pirutka. The results rely on a detailed study of the
third unramified cohomology group of specific products of varieties.
