Magnetic domain walls (DWs) in perpendicularly magnetised thin films are attractive for racetrack memories, but technological progress still requires further reduction of the operationing currents. To efficiently drive these objects by the means of electric current, one has to optimize the damping-like torque which is caused by the spin Hall effect (SHE). This not only requires a high net spin Hall angle but also the presence of a Dzyaloshinskii-Moriya interaction (DMI) to produce magnetic textures sensitive to this type of the torque. In this work, we explore the coexistence and importance of these two phenomena in epitaxial Pt/Co/Pt1-x Au x films in which we control the degree of inversion symmetry-breaking between the two interfaces by varying x. Gold is used as a material with negligible induced magnetic moment and SHE and the interface between Co/Au as a source of a small DMI. We find no current-induced DW motion in the symmetric Pt/Co/Pt (x = 0) trilayer. By fitting a one-dimensional model to the DW velocity as a function of drive current density and in-plane applied field in samples with non-zero values of x, we find that both net DMI strength and spin Hall angle rise monotonically as Au is introduced. They reach values of 0.75 ± 0.05 mJ m-2 and 0.10 ± 0.01, respectively, for Pt/Co/Au (x = 1).