Steenbrink-type vanishing for surfaces in positive characteristic

Let $(X,B)$ be a pair of a normal surface over a perfect field of characteristic $p>0$ and an effective $\mathbb{Q}$-divisor $B$ on $X$. We prove that Steenbrink-type vanishing holds for $(X,B)$ if it is log canonical and $p>5$, or it is $F$-pure. We also show that rational surface singularities satisfying the vanishing are $F$-injective.