We study the problem of assigning doctors to existing, non-operational Primary Health Centers (PHCs). We do this in the presence of clear guidelines on the maximum population that can be served by any PHC, and uncertainties in the availability of the doctors over the planning horizon. We model the problem as a robust capacitated multi-period maximal covering location problem with server uncertainty. Such supply-side uncertainties have not been accounted for in the context of multi-period facility location in the extant literature. We present an MIP formulation of this problem, which turns out to be too difficult for an off-the-shelf solver like CPLEX. We, therefore, present several dominance rules to reduce the size of the model. We further propose a Benders decomposition based solution method with several refinements that exploit the underlying structure of the problem to solve it extremely efficiently. Our computational experiments show one of the variants of our Benders decomposition based method to be on average almost 1000 times faster, compared to the CPLEX MIP solver, for problem instances containing 300 demand nodes and 10 facilities. Further, while the CPLEX MIP solver could not solve most of the instances beyond 300 demand nodes and 10 facilities even after 20 hours, two of our variants of Benders decomposition could solve instances upto the size of 500 demand nodes and 15 facilities in less than 0.5 hour, on average.