  Conditionally on the $abc$ conjecture, we generalize previous work of Clark
and the author to show that a superelliptic curve $C: y^n = f(x)$ of
sufficiently high genus has infinitely many twists violating the Hasse
Principle if and only if $f(x)$ has no $\mathbb Q$-rational roots. We also show
unconditionally that a curve defined by $C: y^{pN}=f(x)$ has infinitely many
twists violating the Hasse Principle over any number field $k$ such that $k$
contains the $p$th roots of unity and $f(x)$ has no $k$-rational roots.
