  Motivated by the weighted Bounded Negativity Conjecture, we prove that all
but finitely many reduced and irreducible curves $C$ on the blow-up of
$\mathbb{P}^2$ at $n$ points satisfy the inequality $C^2 \ge \min
\{-\frac{1}{12} n (C.L +27), -2 \}$, where $L$ is the pull-back of a line. This
partially improves on some result by Laface and Pokora.
