We study the nonlocal Poisson equation with a discontinuous right-hand side f. Whether the solution exhibits discontinuities depends on both the structure of the discontinuity in f and the regularity of the kernel defining the nonlocal operator. For operators with integrable kernels, we show that the solution u inherits jump discontinuities from f. Moreover, the behavior of higher-order derivatives of u near these jumps is governed by the integrability and differentiability properties of the kernel K and its higher derivatives. Numerical approximations of such problems using classical methods--including finite element, finite difference, and spectral methods--typically suffer from reduced accuracy in the presence of discontinuities. To address this challenge, we develop a semi-analytic spectral method that resolves discontinuities with high accuracy. The method is based on transforming the original discontinuous nonlocal equation into a smoother nonlocal problem by exploiting the convolution structure of the operator and symbolic computation techniques. A spectral solver is then applied to the smoothed problem to obtain a highly accurate numerical solution, which is subsequently combined with precomputed analytical functions to reconstruct a semi-analytic solution of the original discontinuous problem.