Subconvexity of Short $k$-Free Exponential Sums in $\mathbb{F}_q[t]$

We extend recent work of the author over $\mathbb{Z}$ into the positive characteristic setting of $\mathbb{F}_q[t]$. In particular, for a polynomial $F \in \mathbb{F}_q[t]$ of degree $N$, let $R_k(α)$ denote the exponential sum over $k$-free polynomials $f$ with $\text{deg}(f-F)0$, we prove essentially tight upper and lower bounds for the $s$-th moment of $R_k(α)$ whenever $K > (\frac{1}{2}+ε)N$, and in even shorter intervals when $s>1+\frac{1}{k}$. As an application of these results, we prove a lower bound of order $q^{\frac{K}{6}}$ for the $L^1$-mean of the Möbius-twisted exponential sum over $\mathbb{F}_q[t]$ whenever $K >(\frac{1}{2}+ε)N$.