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Fix an integer $r\\geq 0$, and let $\\underline{k}=(k_1,\\ldots,k_n)$ be a sequence of positive integers with $k_1+\\ldots+k_n=g-1$. We study $n$-pointed curves $(C,p_1,\\ldots,p_n)$ such that the line bundle $L:=O_C\\left(\\sum_{i=1}^n k_i p_i\\right)$ is a theta-characteristic such that $h^0\\left(C,L\\right)$ is at least $r+1$ and it has the same parity as $r+1$. We prove that they describe a sublocus $\\mathcal{G}^r_g(\\underline{k})$ of $\\mathcal{M}_{g,n}$ having codimension at most $g-1+\\frac{r(r-1)}{2}$. 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