  A pair $(S,C)$ is called a singular $\mathbb{Q}$-homology plane pair if $S$
is a singular projective surface with only quotient singularities having the
same rational homology as $\mathbb{p}^2$ and $C \subset S$ has the same
rational homology as $\mathbb{p}^1$. We will prove results concerning smooth
rational curves on $S$ and the singularities of $S$ such that $\overline
\kappa(S^0)=1$ and $\overline \kappa(S-C) \neq -\infty$. We end with an example
of such pairs.
