  Let $p$ be an odd prime number, and $F$ a nonarchimedean local field of
residual characteristic $p$. We classify the smooth, irreducible, admissible
genuine mod-$p$ representations of the twofold metaplectic cover
$\widetilde{\textrm{Sp}}_{2n}(F)$ of $\textrm{Sp}_{2n}(F)$ in terms of genuine
supercuspidal (equivalently, supersingular) representations of Levi subgroups.
To do so, we use results of Henniart--Vign\'{e}ras as well as new technical
results to adapt Herzig's method to the metaplectic setting. As consequences,
we obtain an irreducibility criterion for principal series representations
generalizing the complete irreducibility of principal series representations in
the rank 1 case, as well as the fact that irreducibility is preserved by
parabolic induction from the cover of the Siegel Levi subgroup.
