Catalyzes the phosphorylation of pseudouridine to pseudouridine 5'-phosphate (PsiMP) (PubMed:31907295). Catalyzes the first step in a pseudouridine degradation pathway (PubMed:31907295). Acts together with the pseudouridine 5'-phosphate glycosidase PUMY in the peroxisome to prevent toxic pseudouridine monophosphate accumulation (PubMed:31907295). ATP + pseudouridine = ADP + H(+) + psi-UMP kcat is 0.23 (sec-1) with pseudouridine as substrate (PubMed:31907295). kcat is 5.28 (sec-1) with pseudouridine as substrate (PubMed:33290549). kcat is 5.80 (sec-1) with ATP as substrate (PubMed:33290549). Forms homodimers. Belongs to the carbohydrate kinase PfkB family.