  We prove Bloch's conjecture for correspondences on powers of complex abelian
varieties, that are "generically defined". As an application we establish
vanishing results for (skew-)symmetric cycles on powers of abelian varieties
and we address a question of Voisin concerning (skew-)symmetric cycles on
powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's
conjecture in the following situation. Let $\gamma$ be a correspondence between
two abelian varieties $A$ and $B$ that can be written as a linear combination
of products of symmetric divisors. Assume that $A$ is isogenous to the product
of an abelian variety of totally real type with the power of an abelian
surface. We show that $\gamma$ satisfies the conclusion of Bloch's conjecture.
A key ingredient consists in establishing a strong form of the generalized
Hodge conjecture for Hodge sub-structures of the cohomology of $A$ that arise
as sub-representations of the Lefschetz group of $A$. As a by-product of our
method, we use a strong form of the generalized Hodge conjecture established
for powers of abelian surfaces to show that every finite-order symplectic
automorphism of a generalized Kummer variety acts as the identity on the
zero-cycles.
