In this paper, we consider the problem of estimating sparse communication channels in orthogonal frequency division multiplexing systems with phase noise and carrier frequency offset. We consider the utilization of the ℓq-norm of the channel as a sparsity-promoting regularization term. The corresponding regularized likelihood cost function is expressed in a Bayesian fashion as a Maximum a Posteriori problem, from which the regularization term is expressed as an a priori distribution for the channel. Given the presence of hidden variables (i.e. the phase noise), the Expectation-Maximization algorithm is utilized. We show that the E-step in the proposed algorithm has a closed-form solution for the channel impulse response. In addition, this approach is an extension of previous work of the authors, which covers the popular Lasso.