The previously reported potential energy surfaces (PESs) of ground-state singlet C-3 and triplet C-4 are here utilized as input for the construction of approximate cluster expansions for larger C-kappa (kappa = 5-10) species. Relying primarily on the double many-body expansion (DMBE) approach, global potentials are obtained by summing the total interaction energies of all atomic subclusters up to four-body terms. The notable capability of the final forms in predicting good estimates of the linear global minima and their thermochemical/structural properties may provide important insights into the structure-determining nature of the (2 + 3 + 4) terms. The main difficulties and virtues in assessing C-kappa's via DMBE theory are analyzed and new prospects given for the construction of global reliable PESs for the target species.