I will ask a slightly more precise question then in the title.
Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 \rangle \oplus \ldots \oplus \langle g_n \rangle$. Is it always possible to find elements $h_1, \ldots, h_n \in G$ and integers $a_1, \ldots, a_n$ such that the following three facts hold
1) $g_i= a_i h_i$, for all $1 \leq i \leq n$,
2) the cyclic subgroups generated by the $h_i$ are in direct sum, $H:=\langle h_1 \rangle \oplus \ldots \langle h_n \rangle$.
3) $H$ is a pure subgroup of $G$?
(recall that a subgroup $H < G$ is pure if for all $h \in H$ and all $n \in \mathbb{N}$, if $h$ is $n$-divisible in $G$, it is also $n$-divisible in $H$).
Added: in this context, pure is a synonim for being a direct summand. In fact, the latter is the relevant property of the question.