Catalyzes the specific phosphorylation of 1,6-anhydro-N-acetylmuramic acid (anhMurNAc) with the simultaneous cleavage of the 1,6-anhydro ring, generating MurNAc-6-P. Is required for the utilization of anhMurNAc either imported from the medium or derived from its own cell wall murein, and thus plays a role in cell wall recycling. 1,6-anhydro-N-acetyl-beta-muramate + ATP + H2O = ADP + H(+) + N-acetyl-D-muramate 6-phosphate Amino-sugar metabolism; 1,6-anhydro-N-acetylmuramate degradation. Cell wall biogenesis; peptidoglycan recycling. Belongs to the anhydro-N-acetylmuramic acid kinase family.