The following is an exercise set by Chris Wendl in his book Holomorphic Curves in Low Dimensions. I'm fairly new to the subject, and wasn't sure how to approach it, so any help with it would be appreciated.
Let $M$ be a smooth, closed, connected, oriented $4$-manifold and $\pi:M\rightarrow \Sigma$ a Lefschetz fibration onto a smooth, closed, connected, oriented $2$-manifold $\Sigma$. Assume that $\pi$ has disconnected fibres. Show that there exists a finite covering $\varphi:\Sigma'\rightarrow \Sigma$ of degree at least $2$, and a factorisation $\pi=\varphi\circ\pi'$, where $\pi':M\rightarrow\Sigma'$ is a Lefschetz fibration with connected fibres.
For completeness we recall that a Lefschetz fibration is a map $\pi$, as in the exercise, with finitely many critical points, such that around each critical point $p$ there exist complex coordinates $(z_1,z_2)$ on $M$ for which $\pi$ has the local form $(z_1,z_2)\mapsto z_1^2+z_2^2$ for some complex chart on $\Sigma $ at $\pi(p)$. Both charts are assumed to be compatible with the orientations on each respective manifold.