Also, would a region of space where light spends more time traversing it become more massive than a similar region of space where light doesn’t spend that much time all else being equal?
One could answer “yes” to your title question. To your text questions the answer is undefined, because the questions themselves start from somewhat incorrect statements.
I say “could/would” because it’s always tricky to interpret someone else’s questions: maybe you have some intuitive pictures in mind, and I may mistranslate them. So take the following with a pinch of salt!
Let’s take the viewpoint of General Relativity. Then “gravitational pull” is really the curvature of spacetime, that makes bodies and radiation gets closer or go apart. According to general relativity, the curvature at a point in space and time is generated by (one could almost say “is the same as”) the amount of energy, energy flow, and pressure (more exactly “stress”) at that point.
Now photons, or more exactly electromagnetic fields, always have an associated energy, energy flow, and pressure. So their energy-force generates curvature.
The equations of general relativity also say that curvature at one point in space and time influences curvature at other points in space and time. So, in principle, a photon passing through a region of space is contributing, owing to its energy-pressure, to the curvature in that region, and therefore to the curvature in nearby regions, which can be experienced as gravitational pull.
But it’s also good to keep in mind that the photon’s contribution to curvature may be extremely small, depending on its energy-pressure.
Regarding your first text question, it isn’t clear what you mean by “massive” region. It’s actually a tricky matter to speak of the “total energy” or “total mass” of a space region – it’s basically a badly defined concept: different people can arrive at different numbers because they use different conventions. There’s a nice discussion about this at this link. Besides that, one must also be very precise in defining “a region of space”. Because what you call a 3D region of space may for another observer be a 2D region (say a surface) moving in time. So to be honest I prefer not to try to overguess your question and give you an answer.
There are many books where you can find this info. A very accessible and publicly available one is General Relativity by Crowell. The standard reference in these matters is Gravitation by Misner, Thorne, Wheeler.
It’s fine not give an answer to the question if the answer is “it depends”. You can always ask for further specifications.
