  For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of
degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both
irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the
large-$q$ limit, we obtain a lower-order term for this count if we consider
non-monic polynomials, which depends on $\Delta$ in a manner which is
consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if
we consider monic polynomials only and $\Delta$ is a scalar. To do this, we use
symmetries of the problem to get for free a small amount of averaging in
$\Delta$. This allows us to obtain additional saving from equidistribution
results for $L$-functions. We do all this in a combinatorial framework that
applies to more general arithmetic functions than the indicator function of
irreducibles, including the M\"{o}bius function and divisor functions.
