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 Q-divisor B on X. We prove that Steenbrink type vanishing holds for $(X,B)$ if it is lc and $p>5$, or it is F-pure. We also show that rational surface singularities satisfying the vanishing are F-injective.