We use resonant inelastic x-ray scattering (RIXS) at the Ir L 3 edge to study the effect of hole doping upon the J eff = 1 2 Mott-insulating state in Sr 2 IrO 4 , via Rh replacement of the Ir site. The spin-wave gap, associated with XY-type spin-exchange anisotropy, collapses with increasing Rh content, prior to the suppression of the Mott-insulating state and in contrast to electron doping via La substitution of the Sr site. At the same time, despite heavy damping, the d-d excitation spectra retain their overall amplitude and dispersion character. A careful study of the spin-wave spectrum reveals that deviations from the J 1 -J 2 -J 3 Heisenberg used to model the pristine system disappear at intermediate doping levels. These findings are interpreted in terms of a modulation of Ir-Ir correlations due to the influence of Rh impurities upon nearby Ir wave functions, even as the single-band J eff = 1 2 model remains valid up to full carrier delocalization. They underline the importance of the transition metal site symmetry when doping pseudospin systems such as Sr 2 IrO 4 .