  Musta\c{t}\u{a} a and Popa introduce the notion of Hodge ideals for an
effective $\mathbb{Q}$-divisor $D$ and prove a vanishing theorem for Hodge
ideals, which generalizes Nadel vanishing for multiplier ideals. However, their
proof needs an extra assumption on the existence of $\ell$-roots of the line
bundle $\mathscr{O}_X(\ell D)$, which is not necessary for Nadel vanishing. In
this paper, we prove that vanishing for Hodge ideals still holds even without
this assumption.
