For enhancing noisy signals, machine-learning based single-channel speech enhancement schemes exploit prior knowledge about typical speech spectral structures. To ensure a good generalization and to meet requirements in terms of computational complexity and memory consumption, certain methods restrict themselves to learning speech spectral envelopes. We refer to these approaches as machine-learning spectral envelope (MLSE)-based approaches. In this paper we show by means of theoretical and experimental analyses that for MLSE-based approaches, super-Gaussian priors allow for a reduction of noise between speech spectral harmonics which is not achievable using Gaussian estimators such as the Wiener filter. For the evaluation, we use a deep neural network (DNN)-based phoneme classifier and a low-rank nonnegative matrix factorization (NMF) framework as examples of MLSE-based approaches. A listening experiment and instrumental measures confirm that while super-Gaussian priors yield only moderate improvements for classic enhancement schemes, for MLSE-based approaches super-Gaussian priors clearly make an important difference and significantly outperform Gaussian priors.