  We prove results on moments of $L$-functions in the function field setting,
where the moment averages are taken over primitive characters of modulus $R$,
where $R$ is a polynomial in $\mathbb{F}_q [T]$. We consider the behaviour as
$\textrm{deg} R \rightarrow \infty$ and the cardinality of the finite field is
fixed. Specifically, we obtain an exact formula for the second moment provided
that $R$ is square-full, and an asymptotic formula for the fourth moment for
any $R$. The fourth moment result is a function field analogue of Heath-Brown's
result in the number field setting, which was subsequently improved by
Soundararajan. Both the second and fourth moment results extend work done by
Tamam in the function field setting who focused on the case where $R$ is prime.
