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. The neuro-cognitive cost of listening with a unilateral CI in multi-talker situations is at present unclear. Here, we simulated listening with a unilateral CI in young, normal-hearing listeners (N = 22) who 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, indicating the behavioural cost of attending to spectrally degraded 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 instead delayed in time: Speech encoding accuracy, modelled on the basis of the sTRF, was significantly enhanced for attended versus ignored intact speech at earlier neural response latencies (<~250 ms). Notably, this attentional enhancement was not absent but delayed for vocoded speech. These findings suggest that attentional selection of unilateral, degraded speech is feasible but comes at the cost of delayed neural separation of competing speech, which might explain listening challenges experienced by unilateral CI users.