Conformal gravity refers to gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations
The simplest theory in this category has the square of the Weyl tensor as the Lagrangian where
The equation of motion upon varying the metric is called the Bach tensor, where
Since these theories lead to fourth-order equations for the fluctuations around a fixed background, they are not manifestly unitary.
Because this equation is a simpler approximation to conformal gravity, m corresponds to the mass of the central source.
So this is a completely different potential from general relativity and not just a small modification.
[3] An alternative approach is to consider the gravitational constant as a symmetry broken scalar field, in which case you would consider a small correction to Newtonian gravity like this (where we consider
By adding a suitable gravitational term to the Standard Model action in curved spacetime, the theory develops a local conformal (Weyl) invariance.
The conformal gauge is fixed by choosing a reference mass scale based on the gravitational constant.
This approach generates the masses for the vector bosons and matter fields similar to the Higgs mechanism without traditional spontaneous symmetry breaking.