  Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of
a prime finite field $\mathbb{F}_p$ by a separated polynomial, that is
$\mathcal{C}: A(y)=B(x)$, where $A(y)$ is an additive polynomial of degree
$p^n$ and the degree $m$ of $B(X)$ is coprime with $p$. Plane curves given by
separated polynomials are well-known and studied in the literature. However
just few informations are known on their automorphism groups. In this paper we
compute the full automorphism group of $\mathcal{C}$ when $m \not\equiv 1 \pmod
{p^n}$ and $B(X)$ has just one root in $K$, that is
$B(X)=b_m(X+b_{m-1}/mb_m)^m$ for some $b_m,b_{m-1} \in K$. Moreover, some
sufficient conditions for the automorphism group of $\mathcal{C}$ to imply that
$B(X)=b_m(X+b_{m-1}/mb_m)^m$ are provided. As a byproduct, the full
automorphism group of the Norm-Trace curve $\mathcal{C}:
x^{(q^r-1)/(q-1)}=y^{q^{r-1}}+y^{q^{r-2}}+\ldots+y$ is computed. Finally, these
results are used to construct multi point AG codes with many automorphisms.
