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Imaging the light racing out from distant sun, as beam of light shoots aways is a circular pattern (spherical actually), remembering that, light comes in photons or packets of energy.
so how come is that we do not see "gaps" in the light coming from distant stars as these "rays" should have gaps that are getting farther apart as distance grows

You are right that single photon detection is a discrete event. But you are under the false assumption that these "rays" are discretely distributed.

Ideally, a photon would have an equal probability of being emitted through any solid angle out of the star. i.e. it is a uniform probability distribution with respect to the solid angle. There aren't single rays that are evenly distributed around the star that the photons travel along.

For a water analogy, it is not like the star is a spherical shower head where photons can only be released from discrete locations. So, even if you might have a different random distribution of photon detection events at different angular locations relative to the star, you will still always see photons (this is neglecting stars that are so far away from us that their light never reaches us due to the expanding universe).

Of course, if you are far enough away you will experience fewer and fewer photons. However this is not limited to certain "rays". This will be true at any angle at a large enough distance.

Very good question. Here is a more QM explanation. It is almost the same as if you would (only for your case) take the Sun as an atom, that is surrounded by an electron field as per QM.

Now the wavefunction of the electron describes the probability distribution of the electron being at a certain position in space around the nucleus.

You would think that the electron can only be at certain discrete number of positions? Well as per QM, the answer is no. In simple words, the electron is at a certain energy level around the nucleus as per QM, but inside that energy level, the electron could be anywhere.

Since the atomic system (and the electron) emits the photons, and the electron could be anywhere (inside the certain energy level as per QM) how would you tell where the electron is at the moment of emission?

So you would imagine that the electron could only take certain fixed positions around the nucleus, and emit the photon from those positions. In reality the electron's position is described by the wavefunction, and it is continuous. Simply said, the electron could be anywhere (inside that certain energy level as per QM).

So in your case if you look at just one single atom, and the atom emits photons from far away, the photons will be continuously distributed. There will be no gaps between the photons.

Now if you look at the Sun, which is made of a whole lot of atoms, you can take it analogously, the photons will be distributed continuously.