Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:

There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)P$ is torsion in $\mathrm{Pic}^0(C)$.

My question is the following. How special is this property? More precisely, fix a genus $g \geq 3$. Are the curves in $\mathcal{M}_g$ with this property a (countable) union of closed subsets?

Some special kinds of curves (e.g. hyperelliptic, or totally branched over $\mathbb{P}^1$) have this property, but it seems to me that it is something quite strong to require. Also, in view of results about torsion packets on curves (see for instance http://people.math.gatech.edu/~mbaker/pdf/packets-final.pdf), the point $P$ above needs to be essentially unique.