Summary: The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by \(\varepsilon^2\), where \(\varepsilon \) is the perturbation parameter, \(\varepsilon \in (0, 1]\). For small values of \(\varepsilon \), an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition \(N^{-1} = o(\varepsilon )\), \(N^{-1} = o(1)\), where \(N + 1\) and \(N_0 + 1\) are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges \(\varepsilon \)-uniformly at the rate \(\mathcal O(N^{-1} \ln N + N_0^{-1})\). Numerical experiments confirm the theoretical results.