  The $t$-$J$-$U$ model of high-$T_c$ copper-oxide superconductors incorporates
both the on-site Coulomb repulsion and kinetic exchange interaction and yields
a semi-quantitative description of the static properties of those materials. We
extend this analysis to dynamic quantities and address collective spin- and
charge excitations in the correlated metallic state of the $t$-$J$-$U$ model.
We employ VWF+$1/\mathcal{N}_f$ approach that combines the variational wave
function (VWF) approach with the expansion in the inverse number of fermionic
flavors ($1/\mathcal{N}_f$). It is shown that the resonant (paramagnon)
contribution to the dynamic magnetic susceptibility remains robust as one
interpolates between the Hubbard- and $t$-$J$-model limits, whereas the
incoherent continuum undergoes substantial renormalization. Energy of the
collective charge mode diminishes as the strong-coupling limit is approached.
We also introduce the concept of effective kinetic exchange interaction that
allows for a unified interpretation of magnetic dynamics in the Hubbard,
$t$-$J$, and $t$-$J$-$U$ models. The results are discussed in the context of
recent resonant inelastic $x$-ray scattering experiments for the high-$T_c$
cuprates.
