<p>In this article, we explore a variant of multi-interval Galerkin techniques utilizing orthogonal Jacobi polynomials and effective Jacobi–Gauss quadrature rules to solve certain Volterra integral models of the second kind that contain weakly singular kernels. The main inspiration behind introducing this multi-interval Galerkin technique is to address the impact of weak singularity of kernels in the models under consideration. This allows us to achieve a piecewise numerical solution within each sub-interval of the computational domain, which is influenced by the value of the weak singularity parameter. Another motivation behind our current research is to directly prove the convergence analysis without relying on any auxiliary problem. This is in contrast to many other research works on global single-step Galerkin Jacobi schemes, where in auxiliary problems certain quadrature rules were overlooked (for simplicity of understanding) during the investigations in the analysis of convergence. Therefore, our approach enhances the understanding of error analysis in terms of exponential convergence rates. We also include several test problems and apply the proposed scheme to them, illustrating the beneficial effects of both p-refinement and h-refinement in our method. This experimental demonstration supports our discussions on error analysis.</p>