Some correlation effects are studied in a narrow-band model with electron-hole asymmetry. The peculiarity of the model is a consideration of the hopping integral caused by electron-electron interaction (correlated hopping). The Green function method is used to derive quasiparticle energy spectrum, widths of the upper and lower Hubbard bands. It is shown that in a series of cases the model leads to the consequences that differ essentially from those of the Hubbard model. The results obtained principally illustrate the model specificity at low temperatures and can be applied to interpret the peculiarities of physical properties of systems with narrow energy bands.