  We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in
which the $i^{th}$ coefficient is either preassigned to be $a_i \in
\mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as
a function field analogue of a recent work of Maynard, which counts integer
primes that do not have specific digits in their base-$q$ expansion. Our work
relates to Pollack's and Ha's work, which count the amount of prime polynomials
with $\ll \sqrt{n}$ and $\ll n$ preassigned coefficients, respectively. Our
result demonstrates how one can prove asymptotics of the number of prime
polynomials with different types of constraints to each coefficient.
