We prove that, for certain indices of $\delta$, there are functions whose Ces\`aro means of order $\delta$ of the Fourier expansion with respect to the polynomials associated with the measure $(1-x)^\alpha (1+x)^\beta+M\Delta_{-1}$, where $\Delta_t$ is the delta function at a point $t$, are divergent almost everywhere on $[-1,1]$. We follow Meaney's paper (2003), where divergent Ces\`aro and Riesz means of Jacobi expansions were proved.