  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 lc and $p>5$, or it
is $F$-pure. We also show that rational surface singularities satisfying the
vanishing are $F$-injective.
