Is it not more a case of space-time being influenced rather than light?

Yes, I would think. Gravity warps space, and light just follows a line in space, which is a curve in space-time. But since gravity can affect light, does light (energy) affect gravity?

Yes. The gravitational field couples to any energy; it doesn't care what the source is. Light carries energy, so the gravitational field responds to it.

More formally, a photon field contributes to the so-called stress-energy tensor,which acts as the 'source' of gravity in the Einstein equations, thus affecting gravity, as you say.

Yeah, in physics if A affects B then B always affects A. This idea was first formalised as Newton's Third Law.