On a family of negative curves

Let $X$ be the blowup of a weighted projective plane at a general point. We study the problem of finite generation of the Cox ring of $X$. Generalizing examples of Srinivasan and Kurano-Nishida, we consider examples of $X$ that contain a negative curve of the class $H-mE$, where $H$ is the class of a divisor pulled back from the weighted projective plane and $E$ is the class of the exceptional curve. For any $m>0$ we construct examples where the Cox ring is finitely generated and examples where it is not.