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The 21 cm transition from neutral Hydrogen promises to be the best observational probe of the Epoch of Reionisation (EoR). This has led to the construction of low-frequency radio interferometric arrays, such as the Hydrogen Epoch of Reionization Array (HERA), aimed at systematically mapping this emission for the first time. Precision calibration, however, is a requirement in 21 cm radio observations. Due to the spatial compactness of HERA, the array is prone to the effects of mutual coupling, which inevitably lead to non-smooth calibration errors that contaminate the data. When unsmooth gains are used in calibration, intrinsically spectrally-smooth foreground emission begins to contaminate the data in a way that can prohibit a clean detection of the cosmological EoR signal. In this paper, we show that the effects of mutual coupling on calibration quality can be reduced by applying custom time-domain filters to the data prior to calibration. We find that more robust calibration solutions are derived when filtering in this way, which reduces the observed foreground power leakage. Specifically, we find a reduction of foreground power leakage by 2 orders of magnitude at k=0.5.

Interferometric experiments designed to detect the highly redshifted 21-cm signal from neutral hydrogen are producing increasingly stringent constraints on the 21-cm power spectrum, but some k-modes remain systematics-dominated. Mutual coupling is a major systematic that must be overcome in order to detect the 21-cm signal, and simulations that reproduce effects seen in the data can guide strategies for mitigating mutual coupling. In this paper, we analyse 12 nights of data from the Hydrogen Epoch of Reionization Array and compare the data against simulations that include a computationally efficient and physically motivated semi-analytic treatment of mutual coupling. We find that simulated coupling features qualitatively agree with coupling features in the data; however, coupling features in the data are brighter than the simulated features, indicating the presence of additional coupling mechanisms not captured by our model. We explore the use of fringe-rate filters as mutual coupling mitigation tools and use our simulations to investigate the effects of mutual coupling on a simulated cosmological 21-cm power spectrum in a \"worst case\" scenario where the foregrounds are particularly bright. We find that mutual coupling contaminates a large portion of the \"EoR Window\", and the contamination is several orders-of-magnitude larger than our simulated cosmic signal across a wide range of cosmological Fourier modes. While our fiducial fringe-rate filtering strategy reduces mutual coupling by roughly a factor of 100 in power, a non-negligible amount of coupling cannot be excised with fringe-rate filters, so more sophisticated mitigation strategies are required.