A well-studied one-dimensional model for the operation of clarifier-thickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. This model also includes a singular feed source. In this paper, the clarifier-thickener model is extended by a singular sink through which material is extracted from the unit. A difficulty is that in contrast to the singular source, the sink term cannot be incorporated into the flux function; rather, the sink is represented by a new non-conservative transport term. The paper is concerned with the well-posedness analysis and numerical methods for the extended model. To simplify the analysis, a reduced problem is formulated, which contains the new sink term of the extended clarifier-thickener model, but not the source term and flux discontinuities that can be handled by existing methods. A definition of entropy solutions, based on Kruzkov-type entropy functions and fluxes, is provided. Jump conditions are derived and uniqueness of the entropy solution is shown. Existence of an entropy solution is shown by proving convergence of a monotone difference scheme. Combining the present analysis for the reduced problem with previous results [Numer. Math. 97 (2004) 25-65] shows that the full extended clarifier-thickener model is well-posed. Two variants of the numerical scheme are introduced. Numerical examples illustrate that all three variants converge to the entropy solution, but introduce different amounts of numerical diffusion.