  We study the electronic structure of HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$
(Hg1223 ; T$_{c}$ = 134 K) using photoemission spectroscopy (PES) and x-ray
absorption spectroscopy (XAS). Resonant valence band PES across the O K-edge
and Cu L-edge identify correlation satellites originating in O 2p and Cu 3d
two-hole final states, respectively. Analyses using the experimental O 2p and
Cu 3d partial density of states show quantitatively different on-site Coulomb
energy for the Cu-site (U$_{dd}$ = 6.5$\pm$0.5 eV) and O-site (U$_{pp}$ =
1.0$\pm$0.5 eV). Cu$_{2}$O$_{7}$-cluster calculations with non-local screening
explain the Cu 2p core level PES and Cu L-edge XAS spectra, confirm the
U$_{dd}$ and U$_{pp}$ values, and provide evidence for the Zhang-Rice singlet
state in Hg1223. In contrast to other hole-doped cuprates and 3d-transition
metal oxides, the present results indicate weakly correlated oxygen holes in
Hg1223.
