Contribution to the emblackening factor from massive gravitons

I have been studying the effects of massive gravitons on the emblackening factor $f(r)$; i.e., given a Reissner-Nordstrom AdS geometry

$$ds^2=L^2 left(frac{dr^2}{f(r)r^2}+frac{-f(r)dt^2+dx^2+dy^2}{r^2} right)quad(textrm{Eqn. } 1) $$
and
$$A(r)=muleft(1-frac{r}{r_0}right)dtquad(textrm{Eqn. } 2) $$
how does $f(r)$ change if we have an additional term in the action proportional to $m^2$, where $m$ is the mass of the graviton and $r_0$ is the black hole event horizon? The additional mass terms from the graviton are discussed here ( "Holography without translational symmetry", by Vegh), and the end result is an emblackening factor given by Eqn. 26:

$$f(r)=1+alpha Ffrac{L m^2}{2}+beta F^2 m^2 r^2-Mr^3+frac{mu^2}{4r_h^2}r^4quad(textrm{Eqn. } 3) $$

where $alpha$ and $beta$ are proportional to the massive gravity term in the action, $M$ is the mass of the black hole, and the massive gravity action is given by $f_{mu nu}=textrm{diag}(0,0,F^2,F^2)$. The derivation of this form appears to make sense to me, but I’ve also noticed a similar term discussed in this work ("Solid Holography and Massive Gravity" by Alberte et. al.), where Eqn. 5.5 yields

$$f(r)=1+frac{mu^2 r^2}{2r_h^2}+r^3int^r dr’ frac{1}{r’^4}V(r’)quad(textrm{Eqn. } 4)$$

where $V(r)$ is the background potential from the massive graviton.

My main question concerns the derivation of Eqn. 4, and how it relates to Eqn. 3. The two solutions look very similar; i.e., it seems as if the effects from the massive gravity term are simply added on to the AdS Schwarzschild solution. Therefore, would it be correct to say that, for some general metric, the addition of a massive gravity background amounts to a constant term multiplying the emblackening factor? If that is the case, how is Eqn. 4 derived? This seems unclear in the reference I linked. Is the additional term proportional to $V(r’)$ due to Einstein’s equations reducing to a set of inhomogeneous differential equations? Any explanation would be greatly appreciated.