We discuss a model in which a nonrelativistic particle can absorb and emit bosonic particles on contact. The bosons have a constant dispersion relation, as in the related Fröhlich polaron model. We determine explicitly the domain of the Hamiltonian for finitely many bosons in terms of singular boundary conditions. The singularities occurring in this model are essentially the same as in the model with quadratic boson dispersion, with simplifications in the formulas highlighting their key features.