We analyze critical phenomena on networks generated as the union of hidden variables models (networks with any desired degree sequence) with arbitrary graphs. The resulting networks are general small-worlds similar to those a` la Watts and Strogatz but with a heterogeneous degree distribution. We prove that the critical behavior (thermal or percolative) remains completely unchanged by the presence of finite loops (or finite clustering). Then, we show that, in large but finite networks, correlations of two given spins may be strong, i.e., approximately power law like, at any temperature. Quite interestingly, if $γ$ is the exponent for the power law distribution of the vertex degree, for $γ\leq 3$ and with or without short-range couplings, such strong correlations persist even in the thermodynamic limit, contradicting the common opinion that in mean-field models correlations always disappear in this limit. Finally, we provide the optimal choice of rewiring under which percolation phenomena in the rewired network are best performed; a natural criterion to reach best communication features, at least in non congested regimes.