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 $R_k(\alpha )$ denote the exponential sum over $k$-free polynomials $f$ with $\text{deg}(f-F)<K$. For all $s>0$, we prove essentially tight upper and lower bounds for the $s$-th moment of $R_k(\alpha )$ whenever $K>(\frac{1}{2}+\epsilon )N$, 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 $K>(\frac{1}{2}+\epsilon )N$.