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, alpha-glutamylthymidine (YdTMP) substitutes for about 20% of the thymidines in the viral DNA, the 80% left are dTMP (PubMed:34522950, PubMed:6792371, PubMed:16789224). These modifications probably prevent degradation of viral genome by the host restriction-modification antiviral defense system (PubMed:34522950, PubMed:21597187). 5-hydroxymethyl-dUMP in DNA + ATP = 5-phosphomethyl-dUMP in DNA + ADP + H(+) The N-terminus contains the 5-hmdU DNA kinase activity. Mutants hmdl and hmd2 are heat-sensitive mutants defective in thymidine hypermodifications with accumulation of 5- PPmdU and 5hmdU, respectively. Belongs to the thymidylate kinase family. 5-hmdU DNA kinase subfamily.