I must disagree that normal superglue doesn't work with the plastic on 4x4x4 cubes. It might well be that some other material like special cements is better. But with my cheap UHU superglue i have successfully repaired pieces on both 4x4x4 and 5x5x5 cubes (rubiks.com ones). The important things to remember is to use enough glue and to clean all contact areas between the 2 pieces to be glued. Make sure that u have the 2 pieces contacting such that the foot will be at the correct angle and position compared with an unbroken piece. Wipe off excess glue with a paper. And work reasonably fast. Even if the superglue is dry within a few minutes don't put the piece back inside ur puzzle for at least 4-5 hours. For gluing 5x5x5 centers it's really important that u get the joint right. Use two edges to make sure they slide freely inside ur glued joined center. If the gap between head and foot is too tight on either side u will most likely experience that the center is likely to rotate on turning an inner layer that contains the center. Or else also it might break again. Should it become too tight u can carefully sand down either the head or the foot until the edges slide freely again
It could also become too tight because there is excess glue left there.