  We consider a family of singular Volterra integral equations that appear in
the study of monotone travelling-wave solutions for a family of
diffusion-convection-reaction equations involving the $p$-Laplacian operator.
Our results extend the ones due to B.\,Gilding for the case $p=2$. The fact
that $p\neq2$ modifies the nature of the singularity in the integral equation,
and introduces the need to develop some new tools for the analysis. The results
for the integral equation are then used to study the existence and properties
of travelling-wave solutions for doubly nonlinear diffusion-reaction equations
in terms of the constitutive functions of the problem.
