Catalyzes the phosphorylation of L-fuculose. Can also phosphorylate, with lower efficiency, D-ribulose, D-xylulose and D-fructose. ATP + L-fuculose = ADP + H(+) + L-fuculose 1-phosphate Optimum pH is 7.8. Carbohydrate degradation; L-fucose degradation; L-lactaldehyde and glycerone phosphate from L-fucose: step 2/3. Belongs to the FGGY kinase family. A 10-residue longer version of this protein (starting with the sequence MLSGYIAGAI) has been identified in strain W3110 (PubMed:9740056). Other evidence, including ribosomal profiling and comparison with other bacterial genera, suggests this protein actually starts on the indicated Met codon (PubMed:25078267). Extended N-terminus. Extended N-terminus. Extended N-terminus.