White light contains the entire visible spectrum including red light, so would it be more effective at charging a battery?
There is a theoretical and a practical issue here (and smaller ones, too).
The theoretical issue is the energy of a photon depends on its frequency (ie color), therefore a blue photon carries more energy than a red one. If you try to charge a battery, then you need more than a photon, so both the color and the intensity of the light/time of irradiation will be a factor.
The practical issue here is that solar cells generally can grab the energy of photons is given energy packages. If photons cannot be harvested at all or only with low efficiency, you cannot charge a battery even upon extended irradiation. As the comment pointed it out, it is not an issue for red light for regular silicon solar cells.