Electric field driven Frèedericksz transition in chiral nematic (cholesteric) planar liquid crystal cell is studied in the presence of flexoelectric effect. An inhomogeneity of electric field and finiteness of anchoring energy are considered. The extra contribution to the electric field induced by flexoelectricity is taken into account as well as the common contribution arising from the applied voltage U. Equilibrium director distribution is obtained in dependence on voltage and flexoelectric coefficients by numerical minimization of the free energy. The threshold voltage was found to decrease with the increase of the flexoelectric coefficients within the range of high flexoelectric coefficients. Thus flexoelectricity promotes the transition. The orientational structure becomes asymmetric about the center of the cell due to flexoelectricity, even in the case of symmetric boundary conditions. The equilibrium structure was also shown to be different for U>0 and U<0. It was found, that for sufficiently high flexoelectric coefficients the director distribution can be described by a simple function. In this case the planar helicoidal structure transfers into a hybrid-aligned one as the voltage increases. This transformation may form the basis for new flexoelectricity-based switching devices. The nonmonotonic behavior of threshold voltage can be observed in the case of small flexoelectric coefficients. An analytical form of the stability conditions was obtained for the planar helicoidal configuration. We have shown that the Frèedericksz transition can be either continuous or discontinuous depending on the material constants and thickness of the liquid crystal cell.