Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p_1,\ldots,p_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "Geometry of Algebraic Curves (vol 2)" by Arbarello et al that the cohomology group $H^1(C,T_C(-p_1-\ldots-p_n))$ parametrizes the infinitesimal deformations of the curve $C$ with $n$ marked points. Here, $T_C$ is the holomorphic tangent bundle of $C$.

Is there a similar interpretation for the cohomology group $H^1(C,T_C(- D))$, where $D$ is an arbitrary effective divisor on $C$ (maybe with the additional assumption that $2-2g-\deg D<0$) with some points occurring with multiplicity (e.g. $D = 2p$, with a point $p\in C$)?