  Let $K$ be the function field of a smooth, irreducible curve defined over
$\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q =
p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in
\overline{K}$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups
$G_n(\beta)=\mathop{\rm{Gal}}(K(f^{-n}(\beta))/K(\beta))$ embed into $[C_q]^n$,
the $n$-fold wreath product of the cyclic group $C_q$. We show that if $f$ is
not isotrivial, then $[[C_q]^\infty:G_\infty(\beta)]<\infty$ unless $\beta$ is
postcritical or periodic. We are also able to prove that if $f_1(x)=x^q+c_1$
and $f_2(x)=x^q+c_2$ are two such distinct polynomials, then the fields
$\bigcup_{n=1}^\infty K(f_1^{-n}(\beta))$ and $\bigcup_{n=1}^\infty
K(f_2^{-n}(\beta))$ are disjoint over a finite extension of $K$.
