  In this note, we show that {\it $\gamma$-hyperelliptic} numerical semigroups
of genus $g \gg \gamma$ satisfy a refinement of a well-known characteristic
weight inequality due to Torres. The refinement arises from substituting the
usual notion of weight by an alternative version, the $K$-weight, which we
previously introduced in the course of our study of unibranch curve
singularities.
