Speech is often degraded by environmental noise or hearing impairment. People can compensate for degradation, but this requires cognitive effort. Previous research has identified frontotemporal networks involved in effortful perception, but materials in these works were also less intelligible, and so it is not clear whether activity reflected effort or intelligibility differences. We used functional magnetic resonance imaging to assess the degree to which spoken sentences were processed under distraction and whether this depended on speech quality even when intelligibility of degraded speech was matched to that of clear speech (close to 100%). On each trial, male and female human participants either attended to a sentence or to a concurrent multiple object tracking (MOT) task that imposed parametric cognitive load. Activity in bilateral anterior insula reflected task demands; during the MOT task, activity increased as cognitive load increased, and during speech listening, activity increased as speech became more degraded. In marked contrast, activity in bilateral anterior temporal cortex was speech selective and gated by attention when speech was degraded. In this region, performance of the MOT task with a trivial load blocked processing of degraded speech, whereas processing of clear speech was unaffected. As load increased, responses to clear speech in these areas declined, consistent with reduced capacity to process it. This result dissociates cognitive control from speech processing; substantially less cognitive control is required to process clear speech than is required to understand even very mildly degraded, 100% intelligible speech. Perceptual and control systems clearly interact dynamically during real-world speech comprehension.SIGNIFICANCE STATEMENT Speech is often perfectly intelligible even when degraded, for example, by background sound, phone transmission, or hearing loss. How does degradation alter cognitive demands? Here, we use fMRI to demonstrate a novel and critical role for cognitive control in the processing of mildly degraded but perfectly intelligible speech. We compare speech that is matched for intelligibility but differs in putative control demands, dissociating cognitive control from speech processing. We also impose a parametric cognitive load during perception, dissociating processes that depend on tasks from those that depend on available capacity. Our findings distinguish between frontal and temporal contributions to speech perception and reveal a hidden cost to processing mildly degraded speech, underscoring the importance of cognitive control for everyday speech comprehension.