Catalyzes the transfer and exchange of Pi from the γ position of ATP to the 5´ -hydroxyl terminus of polynucleotides (double-and single-stranded DNA and RNA) and nucleoside 3´-monophosphates. Polynucleotide Kinase also catalyzes the removal of 3´-phosphoryl groups from 3´-phosphoryl polynucleotides, deoxynucleoside 3´-monophosphates and deoxynucleoside 3´-diphosphates (1).