This work explores a class of nonlinear singularly perturbed first-order delay differential equations with discontinuous source terms. A classical finite-difference scheme is constructed using a suitable piecewise uniform Shishkin mesh. The proposed numerical method, combined with a two-mesh algorithm, is demonstrated to achieve essentially first-order convergence, independent of the perturbation parameter. The delay and discontinuity contribute to the formation of interior layers. The numerical results support the theoretical predictions.