A laser cutting machining apparatus for laser cutting a workpiece includes a first device for generating a machining laser beam, for producing cuts with cut edges in the workpiece, and for partially fine machining a cut edge, and a second device for splitting the machining laser beam into at least two energy intensity ranges. A first energy intensity range for cutting the workpiece has a greater time-integrated radiation energy than a second energy intensity range for fine machining a cut edge. The second device is configured to adjusts at least one element from: a focus diameter of the machining laser beam and/or of at least one of the energy intensity ranges, a focal position of the machining laser beam and/or of at least one of the energy intensity ranges, and a focusing of the machining laser beam and/or of at least one of the energy intensity ranges.