Mixing in droplets is an essential task in a variety of microfluidic systems. Inspired by electrokinetic mixing, electric field-induced hydrodynamic flow inside a charged droplet embedded in an unbounded polyelectrolyte hydrogel is investigated theoretically. In this study, the polyelectrolyte hydrogel is modeled as a soft, and electrically charged porous solid saturated with a salted Newtonian fluid, and the droplet is considered an incompressible Newtonian fluid. The droplet-hydrogel interface is modeled as a surface, which is located at the plane of shear, with the electrostatic potential . The fluid inside the droplet attains a finite velocity owing to hydrodynamic coupling with the electroosmotic flow arising from the droplet and polymer charge. The fluid velocity inside the droplet is linearly proportional to the electroosmotic flow velocity in the charged gel and the electroosmotic flow velocity beyond the electrical double layer of a charged interface. It is found that the polymer boundary condition at the droplet surface and the viscosities of the fluids inside and outside the droplet significantly modulate the interior fluid flow. The ionic strength and the permeability of the polymer network impact the flow differently depending on whether the flow arises from the droplet or polymer charge. Finally, the displacement of a charged droplet embedded in a gel under the influence of an external electric field is undertaken. This work is motivated by experimental attempts, which can register sub-nanometer-scale inclusion displacements in hydrogels, to advance electrical microrheology as a diagnostic tool for probing inclusion-hydrogel interfaces. In the absence of polymer charge, a close connection is found between the electrical response of a charged droplet when it is immobilized in an uncharged incompressible gel and when it is dispersed in a Newtonian electrolyte.