Sugar kinase that catalyzes the ATP-dependent phosphorylation of N-acetylmuramate (MurNAc) and N-acetylglucosamine (GlcNAc) at its C1 hydroxyl group, leading to MurNAc alpha-1P and GlcNAc alpha-1P, respectively (By similarity). Is likely involved in peptidoglycan recycling as part of a cell wall recycling pathway that bypasses de novo biosynthesis of the peptidoglycan precursor UDP-MurNAc (PubMed:23831760). Is able to complement the fosfomycin sensitivity phenotype of a P.putida mutant lacking amgK (PubMed:23831760). ATP + N-acetyl-D-muramate = ADP + H(+) + N-acetyl-alpha-D-muramate 1-phosphate ATP + N-acetyl-D-glucosamine = ADP + H(+) + N-acetyl-alpha-D-glucosamine 1-phosphate Cell wall biogenesis; peptidoglycan recycling. Belongs to the kinase AmgK family.