Let me describe a natural straightforward generalization of Chris Wutrich's counterexample.
Let $B$ be a $g$-dimensional abelian variety over a field $k$ and assume that $End_k(B)$ is a principal ideal domain. Let $A$ be another abelian variety over $k$ that is not $k$-isomorphic to $B$ but $k$-isogenous to it. Then the group $Hom_k(A,B)$ becomes a free $End_k(B)$-module of rank 1. This means that there exists a (generator) isogeny $\lambda:A \to B$ such that every $k$-homomorphism $v: A\to B$ is a composition $u\lambda$ of $\lambda$ and a certain $u \in End_k(B)$. In particular, $ker(v)$ always contains $\ker(\lambda)$. Since $A$ and $B$ are not isomorphic over $k$, the isogeny $\lambda$ is not an isomorphism and therefore $H:=ker(\lambda)\subset A$ is nontrivial but is killed by every $k$-homomorphism from $A$ to $B$.
In order to construct explicit examples (over finite fields) pick any imaginary quadratic field $K$ of discriminant 1 amd let $O$ be the ring of integers in $K$, which is PID. Then for a ``half" of the primes $p$ there exist a finite field $k$ of characteristic $p$ and an ordinary elliptic curve $B$ over $k$ with $End_k(B)=End(B)=O$. Enlarging (if necessary) $k$, we may find an elliptic curve $A$ over $K$ that is not $k$-isomorphic to $A$ but $k$-isogenous to it. For example, if a prime $\ell$ is different from $p$ and inert in $O$ then we may pick a cyclic order $\ell$ subgroup $C$ in $B(k)$ (enlarging $k$ if necessary) and put $A=B/C$. Then the cyclic order $\ell$ subgroup $H=B_{\ell}/C\subset A$ is killed by every homomorphism $A \to B$.