Mollified moments of quadratic Dirichlet $L$-functions over function fields

We compute asymptotic formulae for the mollified first and second moments for the family of quadratic Dirichlet $L$-functions in the function field setting. As an application, we obtain non-vanishing results for the derivatives of the completed $L$-functions $\Lambda(s,\chi_D)$ at the central point $s=1/2$. In particular, we show that the proportion of $\Lambda^{(2k)}(\frac{1}{2},\chi_D) \neq 0$ is $1+O(k^{-2})$ as $k \to \infty$.