This paper considers the impact of general hardware impairments in a multiple-antenna base station and user equipments on the uplink performance. First, the effective channels are analytically derived for distortion-aware receivers when using finite-sized signal constellations. Next, a deep feedforward neural network is designed and trained to estimate the effective channels. Its performance is compared with state-of-the-art distortion-aware and unaware Bayesian linear minimum mean-squared error (LMMSE) estimators. The proposed deep learning approach improves the estimation quality by exploiting impairment characteristics, while LMMSE methods treat distortion as noise.