<p>This paper focuses on the numerical solution of a second-order Volterra integro-differential equation involving delay and neutral types, a class of models underexplored despite their importance in capturing complex dynamical systems with delayed feedback and memory effects. To lay a theoretical foundation for the numerical analysis, stability inequalities for the exact solutions with respect to the right-hand side and initial conditions are first established. On a constrained uniform mesh that aligns all primary discontinuity points with grid nodes, a finite difference scheme is constructed using central differences for the differential terms and a combined composite trapezoidal and midpoint rectangle rule for the Volterra integral term. A key advantage of the proposed scheme is that it can be rearranged into an explicit iterative form, which reduces the computational cost per step and simplifies code implementation. By using different forms of discrete Grönwall’s inequalities, the proposed scheme is proved to achieve second-order convergence in the discrete maximum norm for smooth solutions, with first-order convergence maintained for piecewise smooth solutions. Five numerical examples with diverse data confirm the theoretical accuracy, demonstrating that the convergence orders closely match the theoretical expectation and validating the effectiveness of this method.</p>