  Spatially extended aggregates of dopants are ubiquitous in a plethora of
granular superconductors, forming a droplet network that is very important to
their characterization and to the description of their superconducting
properties. At the same time, one of the most studied classes of
superconducting materials are the high-temperature superconductors, where
special attention is given to the hole-doped cuprates, where the carrier
concentration is controlled by the amount of interstitial oxygen dopants. In
this context, the formation of spatially inhomogeneous aggregates of
interstitial oxygen atoms, in the form of nanosized superpuddles, is not only
relevant, but also a subject of intense recent experimental and theoretical
surveys. Following these efforts, in this work we investigate the consequences
of the presence of networks of inhomogeneously distributed dopant superpuddles
on the superconducting state. Starting from the inhomogeneous extended
disordered background brought by the network of superpuddles, we demonstrate,
with the aid of an effective interaction between electrons mediated by the
local vibrational degrees of freedom of each puddle, that the Cooper pairs
arising from an attractive interaction in an inhomogeneous medium have a finite
center-of-mass momentum, $\mathbf{p}$, that breaks up the Cooper channel.
Furthermore, we derive an analytical expression for the amplitude of the
superconducting gap, $\Delta_{\mathbf{k}}$, in terms of disorder and finite
center-of-mass momentum and show that amplitude fluctuations are induced in the
superconducting state by the presence of the superpuddles, where both the gap
and the critical temperature are reduced by disorder and finite momentum pairs.
Finally, we discuss our findings in the context of synchronized networks of
superconducting oxygen nano-puddles in cuprates and in other granular
superconducting systems.
