No. Light moves on null worldlines, is what
@PeroK said. Photons don’t move it all, at least not the way you’re thinking of ‘move’.
The problem here is that photons aren’t what you’re thinking when you hear phrases like “quantum of light” or “massless particle”. There’s no really good math-free way of saying what a photon is, but here is the best I can come up with:
We have electrical and magnetic fields (and sometimes these fields form traveling waves which we call “light” and which propagate at speed ##c## in a vacuum). We find that whenever these fields interact with matter, the interaction always exchanges energy and momentum in discrete amounts and at a single point (unlike classical wave behavior, where it is spread out across the entire wavefront). When this happens, we say “a photon was detected” at that point.
Now the idea of photons “moving” makes no sense. A light source transfers energy to the electromagnetic field in units of photons. This changes the field, and at some later time the field will exchange energy with whatever the light source is illuminating, again in units of photons. But that’s not a photon moving from one point to another, it’s two separate interactions with the quantized electromagnetic field at two different locations.