Assume the force of gravity is proportional to $1/(R+d)^2$ where $d$ is sufficiently small,
then, how small would $d$ have to be to evade detection by modern experiments?
My thinking is that if $d$ is not zero then it would solve the issue of a singularity when $R$ is 0.
I answered a related question in this earlier answer where I pointed to a 1988 experiment/paper by Mitrofanov et al, where they determined that any deviation from the inverse square law must happen at length scale < 1 mm.
Note that the assumption of their paper was a different form of the potential function -
$$V(r) = -G_\infty\frac{M}{r}\left(1+\alpha e^{-r/r_0}\right)$$
But the basic length scale will be comparable.
