Catalyzes the ATP-dependent phosphorylation of D-threonate to D-threonate 4-phosphate. Can also phosphorylate 4-hydroxy-L-threonine, with lower efficiency. ATP + D-threonate = 4-O-phospho-D-threonate + ADP + H(+) kcat is 41 sec(-1) with D-threonate as substrate. kcat is 28 sec(-1) with 4-hydroxy-L-threonine as substrate. Belongs to the four-carbon acid sugar kinase family.