Quantum dots (QDs) realized by etching or gate structures in a two-dimensional electron system may, under realistic conditions, contain an impurity atom which causes deviations of the electron structure from the ideal case. We have performed ground-state calculations for circular disk-like QDs with a parabolic confinement potential and a repulsive off-center impurity. Besides the distance of the impurity from the dot center we can vary its strength using a Coulomb potential for its shape. The essential effect of the off-center impurity is the breaking of the axial symmetry with the consequence of removing degeneracies in the single-particle spectrum which is related to shell and spin phenomena. We quantify these effects in the ground-state energy and the spin densities by applying spin-density functional theory (SDFT) in the local spin density approximation (LSDA).