Phosphorylates dGMP, dTMP and 5-hydroxymethyl-dCMP while excluding dCMP and dAMP. The phosphorylation of 5-hydroxymethyl-dCMP represents the first step in the replacement of cytosine by hydroxymethylcytosine in new viral DNA genomes. a 2'-deoxyribonucleoside 5'-phosphate + ATP = a 2'-deoxyribonucleoside 5'-diphosphate + ADP Homodimer. Belongs to the dNMP kinase family.