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Let $X$ be a proper curve over $k$ (algebraically closed) of characteristic $p>0$.

When is $H_{fl}^2(X,\mu_n)$ is a finite group?

It's true when $X$ is smooth but are there any more general results?

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    $\begingroup$ If the Picard scheme of $X$ has any addtive subgroups, then the Kummer sequence implies that the group is not finite. If the Picard scheme is semi-abelian, then you can prove finiteness using the vanishing of the Brauer group of any curve over an algebraically closed field (Grothendieck: Groupe de Brauer III, Corollaire 1.2) $\endgroup$
    – naf
    Commented Nov 29, 2018 at 12:31

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