Global $F$-regularity for weak del Pezzo surfaces

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a normal projective surface over $k$ with canonical singularities whose anti-canonical divisor is nef and big. We prove that $X$ is globally $F$-regular except for the following cases: (1) $K_X^2=4$ and $p=2$, (2) $K_X^2=3$ and $p\in \{2,3\}$, (3) $K_X^2=2$ and $p\in \{2,3\}$, (4) $K_X^2=1$ and $p\in \{2,3,5\}$. For each degree $K_X^2$, the assumption of $p$ is optimal.