  A nonzero-temperature technique for strongly correlated electron lattice
systems, combining elements of both variational wave function (VWF) approach
and expansion in the inverse number of fermionic flavors ($1/\mathcal{N}_f$),
is developed. The departure point, VWF method, goes beyond the renormalized
mean-field theory and provides semi-quantitative description of principal
equilibrium properties of high-$T_c$ superconducting cuprates. The developed
here scheme of VWF+$1/\mathcal{N}_f$, in the leading order provides dynamical
spin and charge responses around the VWF solution, generalizing the
weak-coupling spin-fluctuation theory to the regime of strong correlations.
Thermodynamic corrections to the correlated saddle-point state arise
systematically at consecutive orders. Explicitly, VWF+$1/\mathcal{N}_f$ is
applied to evaluate dynamical response functions for the hole-doped Hubbard
model and compared with available determinant quantum-Monte-Carlo data,
yielding a good overall agreement in the regime of coherent collective-mode
dynamics. The emergence of well-defined spin and charge excitations from the
incoherent continua is explicitly demonstrated and a non-monotonic dependence
of the charge-excitation energy on the interaction magnitude is found. The
charge-mode energy saturates slowly when approaching the strong-coupling limit,
which calls for a reevaluation of the $t$-$J$-model approach to the charge
dynamics in favor of more general $t$-$J$-$U$ and $t$-$J$-$U$-$V$ models. The
results are also related to recent inelastic resonant $X$-ray and neutron
scattering experiments for the high-$T_c$ cuprates.
