  Let $\KC_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface
and $C\subset S$ a genus $g$ curve with divisibility $k$ in $\Pic(S)$. In this
article we study the forgetful map $c_g^k:(S,C) \mapsto C$ from $\KC_g ^k$ to
$\M_g$ for $k>1$. First we compute by geometric means the dimension of its
general fibre. This turns out to be interesting only when $S$ is a complete
intersection or a section of a Mukai variety. In the former case we find the
existence of interesting Fano varieties extending $C$ in its canonical
embedding. In the latter case this is related to delicate modular properties of
the Mukai varieties. Next we investigate whether $c_g^k$ dominates the locus in
$\M_g$ of $k$-spin curves with the appropriate number of independent sections.
We are able to do this only when $S$ is a complete intersection, and obtain in
these cases some classification results for spin curves.
