  We propose a systematic approach to the systems of correlated electrons, the
so-called $\mathbf{k}$-DE-GWF method, based on reciprocal-space
($\mathbf{k}$-resolved) diagrammatic expansion of the variational
Gutzwiller-type wave function for parametrized models of correlated fermions.
The present approach, in contrast to either variational Monte-Carlo (VMC), or
the recently developed real-space diagrammatic expansion of the Gutzwiller-type
wave function (direct-space DE-GWF technique), is applicable directly in the
thermodynamic limit and thus is suitable for describing selected singular
features of the wave-vector-dependent quantities. We employ the
$\mathbf{k}$-DE-GWF method to extract the non-analytic part of the two leading
moments of the fermion spectral-density function across the (two-dimensional)
Brillouin zone for the Hubbard model and away from the half-filling. Those
moments are used to evaluate the nodal quasiparticle velocities and their
spectral weights in the correlated superconducting state. The two velocities
determined in that manner exhibit scaling with the electron concentration
qualitatively different from that obtained earlier for the excited states of
the high-$T_c$ cuprates within the projected quasi-particle ansatz, and the
results are in a very good quantitative agreement with experimental data if
interpreted as those characterizing the spectrum below and above the observed
kink. We provide a detailed discussion of the two gaps and two excitation
branches (two velocities) appearing naturally within our DE-GWF approach. The
two separate sets of characteristics distinguish the renormalized quasiparticle
states very close to the Fermi surface from the deeper correlated-state
properties. Also, an enhancement of the $\mathbf{k}$-dependent magnetic
susceptibility is shown to contain a spin-fluctuation contribution within our
language.
