I understand that the Poynting–Robertson effect describes solar radiation-induced drag on certain sized dust particles, causes them to lose orbital angular momentum and eventually fall into the Sun.
But how much mass per year is actually added to the Sun from this effect?
The Sun does not accrete any mass as a result of the Poynting-Robertson (P-R) effect. The P-R effect only drags in relatively small dust particles and these are easily evaporated/sublimated when they get within a few solar radii of the Sun's surface and the atoms/ions are blown away again as part of the solar wind.
Details
As I explained in my answer to The Poynting Robertson Effect, the P-R effect is one of three competing forces on dust particles in the Solar System.
If dust is smaller than $R \sim 10^{-7}$ m then outward radiative forces exceed any gravitational forces or P-R drag, and the particles are blown out of the Solar System at any distance from the Sun. For dust particles a little bigger than this then the P-R effect will cause a drag and spiral-in on a timescale of around $10^4$ years (at the distance of the Earth). This timescale grows linearly with the size of the particle and gets longer than the age of the Solar System once you get to the size of rocks or if they start further away from the Sun.
Thus we expect all dust on size scales of $10^{-6} < R < 0.1$ m to get dragged towards the Sun if it initially orbits at around the distance of the Earth. But since the P-R drag gets weaker as you get further away from the Sun and decreases faster than gravity or radiation forces, the drag timescales all get longer by the square of starting distance and thus the upper limit on dust size that can be dragged towards the Sun in the Solar System lifetime decreases, and will correspondingly be of order of mm at the asteroid belt for instance.
Having got all that out of the way, the conclusion is that dust in the Solar System that meets these size criteria will get dragged towards the Sun. The problem is that the Sun is hot!. When the sub-mm particles get about 0.01-0.02 au (about 1.5-3 million km) from the Sun's photosphere they will just sublimate, their atoms/ions join the outwardly flowing solar wind and never reach the Sun (Pitjeva et al. 2021).
Thus the Sun's mass does not increase at all due to the P-R effect. Pitjeva et al. (2021) do however give an estimate of $10^{-17}$ solar mass/year for the amount of dust falling in towards the Sun as a result of the P-R effect. But I re-emphasize - none of it reaches the solar photosphere.
Even if the Sun gains some mass due to the Poynting-Robertson effect, it most certainly is outweighed by its fusion processes. According to this article: Is the Sun Shrinking?, the Sun looses 4.289x10^12 grams every second to energy. Because of this, the Sun is constantly looing mass. Not to mention the constant solar radiation and wind the Sun is blowing away every second. Poynting-Radiation effects certainly do not outweigh this mass loss of the Sun. It is worth noting that even though the Sun looses a lot of mass due to Energy and solar wind, it is a small portion of its real mass.
