  In this article we consider moduli properties of singular curves on K3
surfaces. Let $\mathcal{B}_g$ denote the stack of primitively polarized K3
surfaces $(X,L)$ of genus $g$ and let $\mathcal{T}^n_{g,k} \to \mathcal{B}_g$
be the stack parametrizing tuples $[(f: C \to X, L)]$ with $f$ an unramified
morphism which is birational onto its image, $C$ a smooth curve of genus
$p(g,k)-n$ and $f_*C \in |kL|$. We show that the forgetful morphism $$\eta \; :
\; \mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ is generically finite on
one component, for all but finitely many values of $p(g,k)-n$. We further study
the Brill--Noether theory of those curves parametrized by the image of $\eta$,
and find a Wahl-type obstruction for a smooth curve with an unordered marking
to have a nodal model on a K3 surface in such a way that the marking is the
divisor over the nodes.
