  We investigate the effect of coupling Anderson localized particles in one
dimension to a system of marginally localized phonons having a symmetry
protected delocalized mode at zero frequency. This situation is naturally
realized for electrons coupled to phonons in a disordered nano-wire as well as
for ultra-cold fermions coupled to phonons of a superfluid in a one dimensional
disordered trap. To determine if the coupled system can be many-body localized
we analyze the phonon-mediated hopping transport for both the weak and strong
coupling regimes. We show that the usual variable-range hopping mechanism
involving a low-order phonon processes is ineffective at low temperature due to
discreteness of the bath at the required energy. Instead, the system
thermalizes through a many-body process involving exchange of a diverging
number $n\propto -\log T$ of phonons in the low temperature limit. This effect
leads to a highly singular prefactor to Mott's well known formula and strongly
suppresses the variable range hopping rate. Finally we comment on possible
implications of this physics in higher dimensional electron-phonon coupled
systems.
