Curvature of a laser beam with M2 parameter greater than 1

Usually a laser beam is characterized with parameter $M^2$ which shows how much does the beam size diverge in comparison to the Gaussian beam.

$w=w_0\sqrt{1+M^2(\frac{z-z_0}{z_R})^2}$

But what in that case is the equation for the beam radius of curvature? Is it the same as for the classical Gaussian beam?

The radius of curvature depends in detail on the beam profile and the aberration present in the beam. Indeed, in an aberrated beam, there is no simple definition of the radius of curvature. One of the problems with the $M$ parameter is that there is no simple relationship between it and the beam details; it is a very blunt characterization. If you want to know what is happenning with your laser, you need to measure its intensity and phase profile. From there, you can calculate the beam's evolution through space with e.g. the Fourier transform method I outline in this answer here. The most powerful and accurate instrument for this job is a point diffraction phase shifting interferometer.