Catalysis of the reaction: a 1-phosphatidyl-1D-myo-inositol 5-phosphate + ATP = a 1-phosphatidyl-1D-myo-inositol 4,5-bisphosphate + ADP + H+.
ATP:1-phosphatidyl-1D-myo-inositol-5-phosphate 4-phosphotransferase activity