  A modified Poisson-Nernst-Planck system in a bounded domain with mixed
Dirichlet-Neumann boundary conditions is analyzed. It describes the
concentrations of ions immersed in a polar solvent and the correlated electric
potential due to the ion--solvent interaction. The concentrations solve
cross-diffusion equations, which are thermodynamically consistent. The
considered mixture is saturated, meaning that the sum of the ion and solvent
concentrations is constant. The correlated electric potential depends
nonlocally on the electric potential and solves the fourth-order Poisson-Fermi
equation. The existence of global bounded weak solutions is proved by using the
boundedness-by-entropy method. The novelty of the paper is the proof of the
weak--strong uniqueness property. In contrast to the existence proof, we
include the solvent concentration in the cross-diffusion system, leading to a
diffusion matrix with nontrivial kernel. Then the proof is based on the
relative entropy method for the extended cross-diffusion system and the
positive definiteness of a related diffusion matrix on a subspace.
