  It has been recently shown that the competition between unscreened Coulomb
and Fr\"{o}hlich electron-phonon interactions can be described in terms of a
short-range spin exchange $J_p$ and an effective on-site interaction
$\tilde{U}$ in the framework of the polaronic $t$-$J_p$-$\tilde{U}$ 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 tunneling
conductance, signaling the presence of normal state pseudogap without any
assumptions on preexisting orders or broken symmetries in the normal state of
the model.
