Let $K$ be a number field $A$ an abelian variety over $K$ and $f: X \to A$ a possibly ramified covering of degree $d$ with $X$ a proper variety. My question is:

> Suppose that $f(X(K)) \neq A(K)$, can ond then always find a finite set of strict subgroups $H_1,\ldots, H_n \subsetneq A(K)$ and a sequence of points $x_1, \ldots, x_n$ such that $f(X(K)) \subseteq \bigcup x_i + H_i \neq A(K)$?

This question is only interesting when $\# A(K) = \infty$. And the general result also easily follows from the case where $X$ is irreducible.

Now there are two cases in which I know the answer to be positive:

 1. When $A$ is an elliptic curve. Because if $X(K) \neq \emptyset$ then either $X$ is an elliptic curve as well in which case $f(X(K))$ is already a coset of a finite index subgroup, or $X$ has genus > 1 in which case $X(K)$ is finite and the statement is trivially true.
 2. $X$ is a subvariety of an abelian variety $B$. Because in this case one can use a result of Faltings theorem to cover the rational points of $X$ by translates of abelian subvarieties of $B$, and the result follows as well.

I personally even have no idea what I should expect the answer to be. Although I am hoping for a positive answer because of some applications I have for this statement.