Singular rational curves with points of nearly-maximal weight

In this article we study rational curves with a unique unibranch genus-$g$ singularity, which is of {\it $\text{ka}$-hyperelliptic} type in the sense of \cite{To}; we focus on the cases $\text{ka}=0$ and $\text{ka}=1$, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.