So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.

p.s. A result of Oort and de Jong proved that a "generic" abelian variety satisfies this property. However it seems generic means outside a countable subfamily in their result.

I'm interested in this question because if we assume all motives over a finite field is semisimple (or just semisimplify it), then by Weil conjecture and Honda-Tate (maybe also Tate's conjecture over finite field), such motive all live in the tensor category of that of abelian varieties, or equivalently that of curves. I am then wondering if there could be any chance that they all actually come from hyperelliptic curves.