  We use resonant inelastic x-ray scattering (RIXS) at the Cu L$_3$ edge to
measure the charge and spin excitations in the "half-stuffed" Cu--O planes of
the cuprate antiferromagnet Ba$_2$Cu$_3$O$_4$Cl$_2$. The RIXS line shape
reveals distinct contributions to the $dd$ excitations from the two
structurally inequivalent Cu sites, which have different out-of-plane
coordinations. The low-energy response exhibits magnetic excitations. We find a
spin-wave branch whose dispersion follows the symmetry of a CuO$_2$ sublattice,
similar to the case of the "fully-stuffed" planes of tetragonal CuO (T-CuO).
Its bandwidth is closer to that of a typical cuprate material, such as
Sr$_2$CuO$_2$Cl$_2$, than it is to that of T-CuO. We interpret this result as
arising from the absence of the effective four-spin inter-sublattice
interactions that act to reduce the bandwidth in T-CuO.
