A major pathway towards understanding complex systems is given by exactly solvable reference systems that contain the essential physics of the system. For the $t-t-U$ Hubbard model, the four-site plaquette is known to have a quantum critical point in the $U-mu$ space where states with electron occupations $N=2, 3, 4$ per plaquette are degenerate [Phys. Rev. B {bf 94}, 125133 (2016)]. We show that such a critical point in the lattice causes an instability in the particle-particle singlet d-wave channel and manifests some of the essential elements of the cuprate superconductivity. For this purpose we design an efficient superperturbation theory -- based on the dual fermion approach -- with the critical plaquette as the reference system. Thus, the perturbation theory already contains the relevant d-wave fluctuations from the beginning via the two-particle correlations of the plaquette. We find that d-wave superconductivity remains a leading instability channel under reasonably broad range of parameters. The next-nearest-neighbour hopping $t$ is shown to play a crucial role in a formation of strongly bound electronic bipolarons whose coherence at lower temperature results in superconductivity. The physics of the pseudogap within the developed picture is also discussed.
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