Psychotic illness has consistently been associated with deficits in cognitive function and reduced white matter integrity in the brain. However, the link between white matter disruptions and deficits in cognitive domains remains poorly understood. We assessed cognitive performance and white matter myelin water fraction (MWF) using multicomponent driven equilibrium single pulse observation of T1 and T2 (mcDESPOT) in recent-onset psychosis patients and age-matched healthy controls (HC). Psychosis patients showed deficits in working memory, phonological and semantic fluency, general intelligence quotient and reduced MWF in the left temporal white matter compared to HC. MWF in the left inferior fronto-occipital fasciculus and inferior longitudinal fasciculus was positively associated with intelligence quotient and verbal fluency in patients, and fully mediated group differences in performance in both phonological and semantic verbal fluency. There was no association between working memory and MWF in the left temporal white matter. Negative symptoms demonstrated a negative association with MWF within the left inferior and superior longitudinal fasciculi. These findings indicate that psychosis-related deficits in distinct cognitive domains, such as verbal fluency and working memory, are not underpinned by a single common dysfunction in white matter connectivity.