Let $\mathbb{F}_{q^n}/\mathbb{F}_q$ be an extension of finite fields.

Is a proper quotient of $\mathbb{F}_{q^n}[x]$ considered as a $\mathbb{F}_q$-algebra always a quotient of $\mathbb{F}_q[x]$ (i.e. no extra generator is necessary)?

I think this comes down to the following. Given an irreducible $r\in \mathbb{F}_q[x]$ of degree $n$ and a positive integer $m$ can we always find $s\in \mathbb{F}_q[x]$ of degree $m$ and $t\in \mathbb{F}_q[x]$ of degree $n$ such that$$r\circ s=t^m$$ as polynomials?