Aiming at exploiting temporal correlations across consecutive time frames in the short-time Fourier transform (STFT) domain, multi-frame algorithms for single-microphone speech enhancement have been proposed, which apply a complex- valued filter to the noisy STFT coefficients. Typically, the multi-frame filter coefficients are either estimated directly using deep neural networks or a certain filter structure is imposed, e.g., the multi-frame minimum variance distortionless response (MFMVDR) filter structure. Recently, it was shown that inte- grating the fully differentiable MFMVDR filter into an end-to- end supervised learning framework employing temporal convolu- tional networks (TCNs) allows for a high estimation accuracy of the required parameters, i.e., the speech inter-frame correlation vector and the interference covariance matrix. In this paper, we investigate different covariance matrix structures, namely Hermitian positive-definite, Hermitian positive-definite Toeplitz, and rank-1. The main differences between the considered matrix structures lie in the number of parameters that need to be estimated by the TCNs as well as the required linear algebra operations, yielding a different computational complexity. For example, when assuming a rank-1 matrix structure, we show that the MFMVDR filter can be written as a linear combination of the TCN outputs, significantly reducing computational complexity. In addition, we consider a covariance matrix estimation procedure based on recursive smoothing, where the smoothing factors are estimated using TCNs. Experimental results on the deep noise suppression challenge dataset show that the estimation procedure using the Hermitian positive-definite matrix structure yields the best performance, closely followed by the rank-1 matrix structure at a much lower complexity. Furthermore, it is shown for the best-performing MFMVDR filters that imposing the MFMVDR filter structure instead of directly estimating the multi-frame filter coefficients slightly but consistently improves the speech enhancement performance.