Gravity can certainly warp and distort the ‘straight-line’ path of a light ray. This Hubble image is of the Einstein Ring LRG 3-757 in which the central massive galaxy has warped the image of a background galaxy into a ring of light. (Credit: ESA/Hubble & NASA)
The speed of light is something measured with a local apparatus in an inertial reference frame, using the same meter stick and clock. A gravitational field has zillions of such ‘locally inertial reference frames’ which are described by freely-falling observers for short intervals of time and small regions of space. In all of these tiny domains, an observer would measure the same velocity for light as guaranteed by special relativity. To ask what the speed of light is over a domain where gravitational forces make a reference frame ‘non-inertial’ and not moving at a constant speed, is an ill-defined question in special relativity. As soon as you try to measure the speed of such an impulse, you would be using a clock and a meter stick which would not be the ‘proper time and space’ intervals for the entire region where the gravitational field exists.
Gravity can affect the speed of light. If you measure the speed over a large enough region that special relativity and its requirement of a flat spacetime is not satisfied. In the presence of curved spacetime, conventional local measurement techniques do not work and so you cannot define the speed of light in exactly the same way that you do under laboratory conditions in ‘flat’ spacetime. In fact, in curved spacetime even the concept of conservation of energy is not easily defined because the curvature of space itself changes the definition. Conservation of energy only works in flat spacetime.