A model of superconductivity in layered high-temperature superconducting cuprates is proposed, based on the extended saddle point singularities in the electron spectrum, weak screening of the Coulomb interaction and phonon-mediated interaction between electrons plus a small short -range repulsion of Hund's or spin-fluctuation origin. This permits to explain the large values of Tc features of the isotope effect, pho:oemission results for the gap and the seemingly conflicting results of Josephson experiments in various geometries.The influence of impurity scattering on Tc, the order parameter at T = 0 and the density of states, including the energy gap, are studied for the same model. It is shown'that the suppression of the gap in the vicinities of the singularities is weak, and the same is true for Tc. Since the order parameter beyond the singularity is still defined by its value in the singular region, it scales with this value and varies only slightly. However, the energy gap in this region, which does not coincide with the order parameter, varies rapidly with impurity concentration, and the gap becomes isotropic in the strong scattering limit. These conclusions agree with preliminary experimental observations.