Negative moments of $L$-functions with small shifts over function fields

We consider negative moments of quadratic Dirichlet $L$--functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $\mathbb{F}_q[x]$, we obtain an asymptotic formula for the $k^{\text{th}}$ shifted negative moment of $L(1/2+\beta,\chi_D)$, in certain ranges of $\beta$ (for example, when roughly $\beta \gg \log g/g$ and $k<1$). We also obtain non-trivial upper bounds for the $k^{\text{th}}$ shifted negative moment when $\log(1/\beta) \ll \log g$. Previously, almost sharp upper bounds were obtained in \cite{ratios} in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon}$.