Momentum is defined as p = mv, where "m" is mass and "v" velocity, and Einstein's famous E=mc2 also applies to photons. It is, therefore, meaningless to speak of a particle having a momentum but not mass. In fact it is just as we used to say (jokingly) "all vector, no force."
The gravitational force (or "pit" in timespace jargon) that is required to detectably affect the photon mass is much greater than anything we can create on earth. The force is much greater than even our sun's gravitational pull.
This is why the discovery of the gravitational lens (predicated by Einstein) proves that photons are subject to gravity and therefore must have mass, otherwise the GR formula E=mc2 does not apply to photons.
So if all this sounds a little bit like "lawyering" it's because it is! Which is why we can say there are is always some doubt, and no absolutes, and no one understands it and there are always convenient "loopholes" or "angles" from which these issues can be handled, sometimes leading to completely different conclusions, yet all of them true!
Let me give you an example in optics. Optical designs can be "optimized" by a series of reiterations on a "subatomic" level, so to say, as minute changes in element specifications. This is where entry-level integral calculus comes in handy, because that's what integration is—minute adjustments where even complex exponential entities can be treated as linear regressions over very short intervals.
A perfect example of this is your eprception of these letters as being curves (such as letters c, s, q, p, etc.) when in fact they are shapes ahcieved by miniscule dots (and the distance between each two dots is simply a straight line) which under a micorscope would lose their smoothness and continuous appearance. In your eyes they are "integrated" into smooth and continuous complex curves.
and the same image reduced 10 times (size=10)
This is no different than looking at your perception of taste on an electron microscope level showing how a molecule of orange juice gets attached to a molecule of your taste buds. Of course, this misses the "bug picture" of what orange juice tastes line on a macroscopic level, i,.e. how we "experience" it, but is not necessarily wrong.
And just as a single brick does not define a house, neither does quatum mechanics describe the world. Just as the photograph of the earth from space does not reflect what it's like to be on earth, neither does the Einstein's theory of relativity describe how we experience gravity (i.e. as spacetime "pits").
In terms of human experience, which is our reality, no science can capture or adequately describe it, and neither can words. But at the same time, we must never fall for the error of believing that what we experience is how things really are, although many do.
For each aspect of reality we must use the most adequate tool ("measuring stick"), and always be aware that it is only one aspect of what is really out there.