On large theta-characteristics with prescribed vanishing

Let $C$ be a smooth projective curve of genus $g\ge 2$. Fix an integer $r\ge 0$, and let $\underset{‾}{k}=(k_1,\dots ,k_n)$ be a sequence of positive integers with $k_1+\dots +k_n=g-1$. We study $n$-pointed curves $(C,p_1,\dots ,p_n)$ such that the line bundle $L:=O_C\left(\sum_{i=1}^n k_i p_i\right)$isatheta-characteristicsuchthat$h^0\left(C,L\right)$ is at least $r+1$ and it has the same parity as $r+1$. We prove that they describe a sublocus ${\mathcal{G}}_g^r(\underset{‾}{k})$ of ${\mathcal{M}}_{g,n}$ having codimension at most $g-1+\frac{r(r-1)}{2}$. Moreover, for any $r\ge 0$, $\underset{‾}{k}$ as above, and $g$ greater than an explicit integer $g(r)$ depending on $r$, we present irreducible components of ${\mathcal{G}}_g^r(\underset{‾}{k})$ attaining the maximal codimension in ${\mathcal{M}}_{g,n}$, so that the bound turns out to be sharp.