Phosphorylates 5-hydroxymethyluracil (5hmdU) into 5-phosphomethyl-2'-deoxyuridine (5- PmdU) on DNA as a step in the pathway leading to thymidine hypermodifications in the viral genome (PubMed:34522950). The phosphate is added internally to the DNA polymer (PubMed:34522950). As a final result of the pathway of hypermodification, 5-aminoethoxy-2'-deoxymethyluridine (5-NeOmdU) substitutes for about 40% of the thymidines in the viral DNA (PubMed:34522950, PubMed:29555775). These modifications probably prevent degradation of viral genome by the host restriction-modification antiviral defense system (PubMed:34522950). 5-hydroxymethyl-dUMP in DNA + ATP = 5-phosphomethyl-dUMP in DNA + ADP + H(+) Belongs to the thymidylate kinase family. 5-hmdU DNA kinase subfamily.