Pseudogap physics in strongly correlated systems is essentially scalc dependent. We generalize the dynamical mean-field theory (DMFT) by introducing into the DMFT equations dependence on the correlation length of pseudogap fluctuations via an additional (momentum-dependent) selfenergy Σk. This self-energy describes nonlocal dynamical correlations induced by short-ranged collective SDW-like antiferromagnetic spin (or CDW-like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with finite correlation length. This generalized DMFT + approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with nearest and next nearest neighbor hopping. The effective single impurity problem is solved by the numerical renormalization group (NRG). Both types of strongly correlated metals, namely (i) the doped Mott insulator and (ii) the case of bandwidth W ≾ U (U is the value of local Coulomb interaction) are considered. Densities of states, spectral functions and ARPES spectra calculated within DMFT + Σk show a pseudogap formation near the Fermi level of the quasiparticle band. We also briefly discuss effects of random impurity scattering. Finally we demonstrate the qualitative picture of Fermi surface «destruction» due to pseudogap fluctuations and formation of «Fermi arcs» which agrees well with ARPES observations.