Does Newtonian mechanics predict the bending of the course of light by objects with mass?

What theory of light do you want to use? There doesn't seem to be any reasonable way to discuss Newtonian mechanics and photons; photons are innately quantum. We can't very reasonably discuss Newtonian gravity and classical electromagnetism, either, since Newton's gravitational law is not Lorentz-invariant, while classical electromagnetism manifestly is.

There was a time when people did calculate the effects of Newtonian gravity on light. Here is a paper from 1804 that does it. One approach is take an object of mass $m$ and initial velocity $\mathbf{v}$ moving past a star with some specified impact parameter. As $m \to 0$, the trajectory of the object converges, so we can take that as a solution for the path of light. (The trajectory it converges to is that of a "test particle" that has no gravitational influence of its own.) I don't know of any significant applications of such a theory.

Today we know that gravity is manifested as the curvature of spacetime, and so affects light, which travels along null geodesics in spacetime. The effects of the gravity of entire galaxies on light traveling through them can be very dramatic. This is what people study in the field of gravitational lensing.

Newton obviously knew that the mass of an object falling under the influence of Earth's gravity has no effect on its acceleration, i.e., all objects accelerate toward Earth at 32 ft/sec/sec regardless of their mass ("weight"). Therefore it follows that an object with no mass, such as a photon, would follow the same rule: it accelerates to earth at 32 ft/sec/sec. (The reason we don't notice this is that photons spend so little time between the object we see and our eyes due to their extreme speed.)

Newton obviously knew this, and logically concluded that photons from distant stars grazing the Sun's limb (edge) would "fall" just a bit towards the Sun as they passed by, resulting in a slightly curved trajectory.

The answer depends on whether light is a particle or a wave. If you imagine light is a particle of some mass travelling at speed c, then you get a Newtonian deflection of light, half of Einstein's value. This is discussed extensively in many places.

But if you think light is a wave, a wave doesn't fall, it only refracts. In order to get a wave of light to bend, you need the frequency of the light to change in different places. In Newtonian gravity, there is no change in the frequency of light waves due to their motion, because there is no coupling of gravity to electromagnetism, and there is no time-dilation which changes the frequency of waves.

So in the wave theory, it was expected that light would be unaffected by gravity. The two theories are reconciled in modern physics, since the time-dilation of gravity is the cause of both the deflection of light (by changing the frequency of light waves) and of the deflection of matter (by changing the frequency of matter waves).