Let $S$ be a complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stablizer $\mathbb Z_m$ along $D$. Suppose now we have a Kahler orbifold $\omega$ metric on $(S,D,m)$ (with metric singularity along $D$). I would like to find a short proof or a reference for the following statement:

Statement. For arbitrary small $\varepsilon$-neighborhood of $D$ in $S$ there is a smooth Kahler metric $\omega_{\varepsilon}$ on $S$ in the same cohomology class as $\omega$ and such that $\omega_{\varepsilon}$ coincides with $\omega$ in $S\setminus U_{\varepsilon}$.

Remark. This statement should hold for all dimensions (assuming the variety $S$ and the divisor $D$ are smooth), if $S$ is $1$-dimensional the statement can be easily proven "by hands".