We investigate the effect of a metal plate on the variable range hopping (VRH) conductivity of a two-dimensional electron-glass (EG) system. The VRH conductivity is known to have a stretched exponentialdependence on temperature, with an exponentpthat depends on the shape of the EG’s single-particle density ofstates (DOS). For constant DOSp=13and for linear DOSp=12, also known as Mott’s and Efros-Shklovskii’sVRH, respectively. The presence of the plate causes two effects on the EG system, static and dynamic. Thewell-known static effect accounts for the additional screening of the Coulomb repulsion in the EG and forthe partial filling of the Coulomb gap in the DOS. This in turn causes an increase of the conductivity at verylow temperatures. Here, we investigate the complementary dynamical effect, which is related to the polaronicphenomena. Our main result is the dynamical suppression of the standard phonon-assisted hopping and, thus,suppression of the conductivity in a much wider range of temperatures as compared to the low-temperature staticeffect. The relation to experiments is discussed.