Conformal blocks from Wilson lines with loop corrections

We compute the conformal blocks of the Virasoro minimal model or its W$_N$ extension with large central charge from Wilson line networks in a Chern-Simons theory including loop corrections. In our previous work, we offered a prescription to regularize divergences from loops attached to Wilson lines. In this paper, we generalize our method with the prescription by dealing with more general operators for $N=3$ and apply it to the identity W$_3$ block. We further compute general light-light blocks and heavy-light correlators for $N=2$ with the Wilson line method and compare the results with known ones obtained using a different prescription. We briefly discuss general W$_3$ blocks.