  Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the
finite field of $q$ elements, and let $p$ be the characteristic of
$\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$
with the property that for all $s \geq t_0$, one has the expected asymptotic
formula in Waring's problem over $\mathbb{F}_q[t]$ concerning sums of $s$
$k$-th powers of polynomials in $\mathbb{F}_q[t]$. For each $k$ not divisible
by $p$, we derive a minor arc bound from Vinogradov-type estimates, and obtain
bounds on $\widetilde{G}_q(k)$ that are quadratic in $k$, in fact linear in $k$
in some special cases, in contrast to the bounds that are exponential in $k$
available only when $k < p$. We also obtain estimates related to the slim
exceptional sets associated to the asymptotic formula.
