Suppose that $Z$ is a finite wedge of some spheres containing circles. Also, let there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume there exists a map $h:X\to Y$ which induces an isomorphism on fundamental groups, where $X$ is a finite wedge of circles.

Is $\pi_2 (M_{h},X)$ a projective $\mathbb{Z}\pi_1 (X)$-module?

Here $M_h =\frac{X\times I\sqcup Y}{(x,1)\sim h(x)}$ denotes the mapping cylinder of $h$.