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Classical spin ices such as Ho₂Ti₂O₇ have generated intense interest, both theoretical and experimental, in the last decade. Despite the success of magnetic monopole models, however, a full understanding of the energetics and relaxation rates in these materials has remained elusive. Additionally, several recent studies have demonstrated that defects including oxygen vacancies and stuffed spins (i.e. additional spins from Ho atoms occupying Ti sites) must be accounted for in understanding spin ice magnetic dynamics. We use a scanning SQUID microscope to measure the temperature dependence of the magnetic flux noise spectrum in three regions with different defect densities, taken over two samples from a single Ho₂Ti₂O₇ crystal. We find qualitative deviations from Arrhenius behavior, including evidence of screening at low frequencies and high temperatures, and compare these observations to expectations from monopole and defect models. Magnetic flux spectroscopy is found to be a powerful tool for studying systems with complex magnetic dynamics.