We show that an electro-osmotic flow near the slippery hydrophobic surface depends strongly on the mobility of surface charges, which are balanced by counterions of the electrostatic diffuse layer. For a hydrophobic surface with immobile charges, the fluid transport is considerably amplified by the existence of a hydrodynamic slippage. In contrast, near the hydrophobic surface with mobile adsorbed charges, it is also controlled by an additional electric force, which increases the shear stress at the slipping interface. To account for this, we formulate electrohydrodynamic boundary conditions at the slipping interface, which should be applied to quantify electro-osmotic flows instead of hydrodynamic boundary conditions. Our theoretical predictions are fully supported by dissipative particle dynamics simulations with explicit charges. These results lead to a new interpretation of zeta potential of hydrophobic surfaces.