Hearing loss is often asymmetric, such that hearing thresholds differ substantially between the two ears. The extreme case of such asymmetric hearing is single-sided deafness. A unilateral cochlear implant (CI) on the more severely impaired ear is an effective treatment to restore hearing. It is unclear how unilateral acoustic degradation and spatial attention to one sound source in multi-talker situations interact. Here, we simulated key features of listening with a unilateral CI: Young, normal-hearing listeners were presented with 8-band noise-vocoded speech to one ear and intact speech to the other ear. Neural responses were recorded in the electroencephalogram (EEG) to obtain the spectro-temporal response function (sTRF) to speech. Listeners made more mistakes when answering questions about vocoded (versus intact) attended speech. At the neural level, we asked how unilateral acoustic degradation would impact the attention-induced amplification of tracking target versus distracting speech. Interestingly, unilateral degradation did not per se reduce the attention-induced amplification but delayed it in time by approximately 100 ms. These findings suggest that attentional selection of unilateral, degraded speech is feasible, but induces delayed neural separation of competing speech, which might explain some of the listening challenges experienced by unilateral CI users.