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The following texts unwittingly suggest that the answer to the question in the title is "yes":

Richard Feynman: "I want to emphasize that light comes in this form - particles. It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you probably learned something about light behaving like waves. I'm telling you the way it does behave - like particles. You might say that it's just the photomultiplier that detects light as particles, but no, every instrument that has been designed to be sensitive enough to detect weak light has always ended up discovering the same thing: light is made of particles." QED: The Strange Theory of Light and Matter p. 15 https://www.amazon.com/QED-Strange-Theory-Light-Matter/dp/0691024170

"Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether." Banesh Hoffmann, Relativity and Its Roots, p.92 https://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/dp/0486406768

"Emission theory, also called Emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. [...] The name most often associated with emission theory is Isaac Newton. In his corpuscular theory Newton visualized light "corpuscles" being thrown off from hot bodies at a nominal speed of c with respect to the emitting object, and obeying the usual laws of Newtonian mechanics, and we then expect light to be moving towards us with a speed that is offset by the speed of the distant emitter (c ± v)." https://en.wikipedia.org/wiki/Emission_theory

"We conclude, therefore, that A BEAM OF LIGHT WILL ACCELERATE IN A GRAVITATIONAL FIELD AS DO OBJECTS WITH REST MASS. For example, near the surface of Earth light will fall with acceleration 9.8 m/s^2." http://web.pdx.edu/~pmoeck/books/Tipler_Llewellyn.pdf

Albert Einstein Institute: "...you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. [...] The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University" http://www.einstein-online.info/spotlights/redshift_white_dwarfs.html

R. V. Pound and J. L. Snider, Effect of Gravity on Gamma Radiation: "It is not our purpose here to enter into the many-sided discussion of the relationship between the effect under study and general relativity or energy conservation. It is to be noted that no strictly relativistic concepts are involved and the description of the effect as an "apparent weight" of photons is suggestive. The velocity difference predicted is identical to that which a material object would acquire in free fall for a time equal to the time of flight." http://virgo.lal.in2p3.fr/NPAC/relativite_fichiers/pound.pdf

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No, you have simply misinterpreted the texts.

Richard Feynman: "I want to emphasize that light comes in this form - particles. ...

He does not mention anything about the speed of the particles. There is no hint of your claimed suggestion.

Moreover, if light consists of particles, ...

You have already previously asked about this one and it has already been explained to you here: What Did the Michelson-Morley Experiment Say about the Speed of Light?

Emission theory, also called Emitter theory or ballistic theory of light ...

Quote taken out of context. The quoted article immediately shows how the theory has been refuted.

We conclude, therefore, that A BEAM OF LIGHT WILL ACCELERATE IN A GRAVITATIONAL FIELD ...

Also taken out of context. In context the acceleration is repeatedly described as a "deflection", meaning that the acceleration changes the direction. There is no statement in there supporting the claim that the speed changes.

you do not need general relativity to derive the correct prediction for the gravitational redshift. ...

This quote is discussing gravitational redshift, not the speed of light.

It is not our purpose here to enter into ...

The velocity difference described in that quote is the velocity difference of the source, not of the light. Nowhere in any of the derivations nor any of the discussions does this source ever claim that the speed of light is altered.

In short, you have misunderstood each of these references, either by taking them out of context or by misapplying their actual statements.

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The answer to your questions is No. The speed of light does not vary with the presence or absence of Gravity. The speed of light is always constant and the value is c.

Now because of the gravity, the space bends which make light to take a curve path instead of straight line. Hence it seems to influence light to travel bit slower but all electromagnetic waves traverse with same speed.

Hope this answers your question.

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    $\begingroup$ "Hence it seems to influence light to travel bit slower but all electromagnetic waves traverse with same speed." That "seems" contradictory. How do you reconcile "light to travel ... slower" with "all ... same speed."?? $\endgroup$
    – Bill N
    Jan 25, 2021 at 22:47
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In Newtonian physics, we only have Galilean relativity. All velocities(of particles) transform with different inertial frames moving with different velocities with respect to each other. So the answer to your question is yes.

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