The speech enhancement task usually consists of removing additive noise or reverberation that partially mask spoken utterances, affecting their intelligibility. However, little attention is drawn to other, perhaps more aggressive signal distortions like clipping, chunk elimination, or frequency-band removal. Such distortions can have a large impact not only on intelligibility, but also on naturalness or even speaker identity, and require of careful signal reconstruction. In this work, we give full consideration to this generalized speech enhancement task, and show it can be tackled with a time-domain generative adversarial network (GAN). In particular, we extend a previous GAN-based speech enhancement system to deal with mixtures of four types of aggressive distortions. Firstly, we propose the addition of an adversarial acoustic regression loss that promotes a richer feature extraction at the discriminator. Secondly, we also make use of a two-step adversarial training schedule, acting as a warm up-and-fine-tune sequence. Both objective and subjective evaluations show that these two additions bring improved speech reconstructions that better match the original speaker identity and naturalness.