It has been recently shown that the competition between unscreened Coulomb and Frohlich electron-phonon interactions can be described in terms of a short-range spin exchange J(p) and an effective on-site interaction (U) over tilde in the framework of the polaronic t-J(p)-(U) over tilde model. This model, that provides an explanation for high-temperature superconductivity in terms of Bose-Einstein condensation (BEC) of small and light bipolarons, is now studied as a charged Bose-Fermi mixture. Within this approximation, we show that a gap between bipolaron and unpaired polaron bands results in a strong suppression of low-temperature spin susceptibility, specific heat and tunnelling conductance, signalling the presence of a pseudogap regime in the normal state without any assumptions on pre-existing orders or broken symmetries in the normal state of the model. Copyright (C) EPLA, 20121003