Does General Relativity allow a reduction in the strength of gravity?

Any theory that extends general relativity must reduce to GR in some limit, and must (of course) agree with experiment. Probably the simplest such extension is Brans-Dicke gravity, which corresponds to replacing the gravitational constant by a scalar field on spacetime: $frac1Gtophi(x)$. Leaving $phi$ as non-dynamical leaves too many degrees of freedom that we need to put in by hand (the values of $phi$ at every point in spacetime) - while this is a well-defined theory, it is not very predictive.

We remedy this by giving a kinetic term for $phi$ in the Lagrangian, which when appropriately dimensionalised, has a constant prefactor $omega$. This field now couples to the Ricci scalar in the Lagrangian, and so has an equation of motion which is determined by the geometry and hence the energy-momentum content. Note that the only free parameter in the theory is $omega$, which must be constrained by experimental bounds. Current observational evidence suggests $omega>40,000$ - for comparison, general relativity is heuristically reobtained in the $omegatoinfty$ limit.

In the new "Einstein equations" relating $R_{munu}$ and $T_{munu}$, you can interpret $phi$ and its derivatives either as constituting a "generalised mass" or a "generalised spacetime curvature" - either way, in the Newtonian limit, it is manifest that this is equivalent to a changing gravitational coupling (even though the Einstein equations themselves have become rather more complicated). This means that regions where $phi$ is large have a low gravitational strength and vice versa.

Here's the equation of motion for $phi(x)$:

$$ squarephi=frac{8pi}{2omega+3}T_alpha^alpha $$

which is a wave equation for massless $phi$, sourced by the trace of the energy-momentum tensor. This means that $phi$-waves propagate at the speed of light away from regions where $T_alpha^alpha>0$.

With the reasonable additional assumption that $phi$ approaches a constant value $tildephisimfrac1G$ asymptotically, we can expand around this value in powers of $frac{1}{omega}$. Provided that the trace of the energy-momentum tensor does not vanish identically, this means that $phisimtildephi+O(1/omega)$ and

$$ R_{munu}-frac12 Rg_{munu}=frac{8pi}{phi}T_{munu}+O(1/omega) $$

If you imagine a single $phi$-wave propagating through space, regions where $phi(x)$ has a crest have below-average gravity while regions in a trough have above-average gravity.

Since, as mentioned, $omega>40,000$, we would expect only very small variations in the gravitational strength. However, regions dominated by electromagnetic energy-momentum have $T_alpha^alpha=0$ and so allow $phi$-waves to travel through them unimpeded, as in a vacuum (since the equation of motion for $phi$ reduces to the free wave equation) - this is what is alluded to in their "long-range nature". So due to their long-range nature and luminal travel, it is possible that many such waves, generated from various sources and different spacetime locations superpose to reduce $G$ in a region (bearing in mind that the paths of these fluctuations are themselves affected by the spacetime geometry, and so lens naturally into a dense region).

This certainly isn't going to resolve the singularity problem (indeed, if any theory that agrees with experiment could, you would see a much larger uptake of it), but it does provide a mechanism to reduce the gravitational strength in regions of high density. Pertinent to your cosmological case is the possibility that $omega$ itself is not a constant, and has stabilised at a high value over time - thus allowing $phi$-waves of greater magnitude, at that epoch creating regions of highly varying gravitational strength.