We propose a second-order version of the resummed thermodynamic perturbation theory for patchy colloidal models with arbitrary number of multiply bondable patches. The model is represented by the hard-sphere fluid system with several attractive patches on the surface and resummation is carried out to account for blocking effects, i.e., when the bonding of a particle restricts (blocks) its ability to bond with other particles. The theory represents an extension of the earlier proposed first order resummed thermodynamic perturbation theory for central force associating potential and takes into account formation of the rings of the particles. In the limiting case of singly bondable patches (total blockage), the theory reduces to Wertheim thermodynamic perturbation theory for associating fluids. Closed-form expressions for the Helmholtz free energy, pressure, internal energy, and chemical potential of the model with an arbitrary number of equivalent doubly bondable patches are derived. Predictions of the theory for the model with two patches appears to be in a very good agreement with predictions of new NVT and NPT Monte Carlo simulations, including the region of strong association.