For $f(x) \in \mathcal{F}_{q^r}[x]$, how to find all $p(x)f(x)$ s.t. its coefficients are $\mathcal{F}_q$-linear over coefficients of $f(x)$?

Question

I want all the coefficients of $p(x)f(x) = \sum_{i=0}^{n-1} b_i x^i$ to be the $\mathcal{F}_q$-linear combinations of coefficients of $f(x) = \sum_{i=0}^{m-1} a_i x^i$, $m < n$. In another words, each $b_i = \sum_{j=0}^{m-1} c_{i,j} a_j$, where each $c_{i,j} \in \mathcal{F}_q$.

Mostly, I am interesting how to find the non-trivial $p(x)$ having the coefficients from $\mathcal{F}_{q^n}$ rather from $\mathcal{F}_q$.