Blocks for Extending a Primitive System in a Free Group

Question

Let $F$ be a free group of rank at least two, $A$ free a factor of $F,$ and $x$ a primitive element in $F \setminus A.$ Suppose that for some/every basis $\mathcal A$ of $A,$ the set $\mathcal A \cup \{x\}$ is not a primitive system in $F$ (does not generate a free factor of $F$). What can be said about the element $x,$ then? Judging by the similar result on free abelian groups, we can say that $$ x \equiv a z^m\, (\mathrm{mod} [F,F]) $$ for some natural number $m > 1,$ $a \in A$ and some primitive $z \in F$ from a free factor of $F$ freely complementing $A.$ But the free groups situation seems to be more complicated. I would be grateful for further explanations, or references.