Distribution of the trace of Frobenius on average for rank 2 Drinfeld modules

Let $q$ be an odd prime power, $a \in \mathbb{F}_q[T]$ and $u \in \mathbb{F}_q^*$. Provided $q \geq 17$, we compute the average number of primes $p$ for which the characteristic polynomial of the Frobenius at $p$ is $X^2 - aX + up$ over a family of rank 2 Drinfeld $ \mathbb{F}_q[T]$-modules. Our results give asymptotic formulas in the $x$-limit.