The purpose of this randomized, double-blind study is to investigate the efficacy and safety of mepolizumab (300 milligram [mg] administered subcutaneously [SC] every 4 weeks) compared with placebo over a 52-week study treatment period in subjects with relapsing or refractory Eosinophilic Granulomatosis with Polyangiitis (EGPA) receiving standard of care therapy including background corticosteroid therapy with or without immunosuppressive therapy. During the treatment period, in accordance with standard of care, corticosteroid dose will be tapered. The key outcomes in the study focus on evaluation of clinical remission, defined as Birmingham Vasculitis Activity Score (BVAS)=0 with a corticosteroid dose of <=4 mg/day prednisolone/prednisone, reduction in disease relapse and reduction in corticosteroid requirement.