Let $L_1$, $\cdots$, $L_k$ be homogenous linear forms in three variables $z_0$, $z_1$, $z_2$ defining $k$ lines in $\mathbb{P}_{2}$. Consider the abelian extension $K ((L_2/L_1)^{1/n}, \cdots, (L_k/L_1)^{1/n}) \supset K:= \mathbb{C}(z_{1}/z_{0}, z_{2}/z_{0})$ with group $G:= (\mathbb{Z}/n\mathbb{Z})^{k-1}$. According to the book Barth-Hulek-Peters-Vandeven (Compact Complex Surfaces, page 240-42), such extension corresponds to a covering $f: X \rightarrow \tilde{\mathbb{P}_{2}}$. Let $P$ be one of the points where at least three lines in $\mathbb{P}_{2}$ meet. Then the lines through point $P$ defines a fibration. I hope to understand the fibration structure. What are the singular fibers and what is the genus of this fibration? If it is any helpful, you may consider the configurations $A_{1}(6)$ (complete quadrangle). Any reference on this is greatly appreciated.

The answer depends on the integer $n$, on the number $k$ of lines and on the configuration of lines. In section 2.5 of this paper

M. Mendes Lopes, R. Pardini, The geography of irregular surfaces, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ. 59, Cambridge Univ. Press (2012), 349–378. arXiv:0909.5195

you can find a computation for $n=5$ and $A_1(6)$.