A central force does not perform work only if the motion is tangential. When the particle moves radially, it has a component of the speed that is not tangent. The cetripetal force and the velocity are no longer paralell so there the centripetal force actually does work on the system. The work is actually $W=\int_{r_i}^{r_f}F_{centripeta}(r) dr$
NOTE: for circular motion you usually use a non-inetrial rotating system where the object is at rest. This introduces a radial pseudoforce, the cetrifugal force. But if the system of reference does not move with the object, such as when there is radial motion, you need to take into account two additional pseudoforces.
One is the Euler force: the fictitious tangential force that is felt in reaction to any radial acceleration.
The other is the Coriolis force, which creates a deflection of moving objects when the motion is described relative to a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right.