We calculate a low-frequency surface impedance of a dirty, -wave superconductor with an imperfect surface incorporating either a thin layer with a reduced pairing constant or a thin, proximity-coupled normal layer. Such structures model realistic surfaces of superconducting materials which can contain oxide layers, absorbed impurities, or nonstoichiometric composition. We solved the Usadel equations self-consistently and obtained spatial distributions of the order parameter and the quasiparticle density of states which then were used to calculate a low-frequency surface resistance and the magnetic penetration depth as functions of temperature in the limit of local London electrodynamics. It is shown that the imperfect surface in a single-band -wave superconductor results in a nonexponential temperature dependence of at which can mimic the behavior of multiband or -wave superconductors. The imperfect surface and the broadening of the gap peaks in the quasiparticle density of states in the bulk give rise to a weakly temperature-dependent residual surface resistance. We show that the surface resistance can be optimized and even reduced below its value for an ideal surface by engineering at the surface using pair-breaking mechanisms, particularly by incorporating a small density of magnetic impurities or by tuning the thickness and conductivity of the normal layer and its contact resistance. The results of this work address the limit of in superconductors at , and the ways of engineering the optimal density of states by surface nanostructuring and impurities to reduce losses in superconducting microresonators, thin-film strip lines, and radio-frequency cavities for particle accelerators.