  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 exponential dependence on
temperature, with an exponent $p$ that depends on the shape of the EG's single
particle density of states (DOS). For constant DOS $p=1/3$ and for linear DOS
$p=1/2$, also known as Mott's and Efros-Shklovskii's VRH respectively. The
presence of the plate causes two effects on the EG system, static and dynamic.
The well known static effect accounts for the additional screening of the
Coulomb repulsion in the EG and for the partial filling of the Coulomb gap in
the DOS. This in turn causes an increase of the conductivity at very low
temperatures. Here we investigate the complementary dynamical effect, which is
related to the polaronic phenomena. 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 static effect. The relation to experiments is discussed.
