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It is well known that supermassive black holes in the centers of galaxies are capable of accelerating charged particles to very high energies. In many cases, the particle acceleration by black holes occurs electromagnetically through an electric field induced by the source. In such scenarios, the accelerated particles radiate electromagnetic waves, leading to the appearance of the backreaction force, which can considerably change the dynamics, especially, if the particles are relativistic. The effect of the radiation reaction force due to accelerating electric field of the central body in curved spacetime has not been considered previously. We study the dynamics of radiating charged particles in the field of the Schwarzschild black hole in the presence of an electric field associated with a small central charge of negligible gravitational influence. We use the DeWitt-Brehme equation and discuss the effect of the self-force, also known as the tail term, within the given approach. We also study the pure effect of the self-force to calculate the radiative deceleration of radially moving charged particles. In the case of bounded orbits, we find that the radiation reaction force can stabilize and circularize the orbits of oscillating charged particles by suppressing the oscillations or causing the particles to spiral down into the black hole depending on the sign of the electrostatic interaction. In all cases, we calculate the energy losses and exact trajectories of charged particles for different values and signs of electric charge.