On the mean value of the functions related to the divisor function on the ring of polynomials over a finite field

Let ${\mathbb{F}}_q[T]$, be the ring of polynomials over a finite field ${\mathbb{F}}_q$. Let $g:{\mathbb{F}}_q[T]\to \mathbb{R}$ be a multiplicative function such that for any irreducible polynomial $P$ over ${\mathbb{F}}_q$ and any $k\ge 1$, the equality $d_k=g(P^k)$ holds for some arbitrary sequence of reals $\{d_k{\}}_{k=1}^{\mathrm{\infty }}$. In this paper, we get an explicit formula for the sum $$T(N)=\sum _{\begin{array}{c}\mathrm{deg}F=NF\text{ is monic}\end{array}}g(F),$$ and also derive different asymptotics when this sum in cases of $ q \to \infty; \ q \to \infty, \ N \to \infty; \ q ^ N \to \infty $.