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 involved in peptidoglycan recycling as part of a cell wall recycling pathway that bypasses de novo biosynthesis of the peptidoglycan precursor UDP-MurNAc (PubMed:24819062). Plays a role in intrinsic resistance to fosfomycin, which targets the de novo synthesis of UDP-MurNAc (PubMed:24819062). 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. Cells lacking this gene accumulate MurNAc. Deletion of this gene increases fosfomycin sensitivity. Growth rate is not affected. Belongs to the kinase AmgK family.