We study the impact of non-radiative defects on Auger recombination in c-plane InGaN/GaN single quantum wells (SQWs) in the efficiency droop regime using high injection time-resolved photoluminescence. The defect density in the SQW is controlled by tuning the thickness of an InAlN underlayer. When the defect density is increased, apart from Shockley-Read-Hall (SRH) and standard Auger recombination, introducing an extra defect-assisted Auger process is required to reconcile the discrepancy observed between the usual ABC model and experimental data. We derive a linear dependence between the SRH coefficient and the bimolecular defect-assisted Auger coefficient, which suggests that the generated defects can act as scattering centers responsible for indirect Auger processes. In particular, in defective SQWs, the defect-assisted Auger recombination rate can exceed the radiative one. Our results further suggest that the defect-assisted Auger recombination is expected to be all the more critical in green to red III-nitride light-emitting diodes due to their reduced radiative rate.