  For a molecular cloud clump to form stars some transport of magnetic flux is
required from the denser, inner regions to the outer regions of the cloud,
otherwise this can prevent the collapse. Fast magnetic reconnection which takes
place in the presence of turbulence can induce a process of reconnection
diffusion (RD). Extending earlier numerical studies of reconnection diffusion
in cylindrical clouds, we consider more realistic clouds with spherical
gravitational potentials and also account for the effects of the gas
self-gravity. We demonstrate that within our setup RD is efficient. We have
also identified the conditions under which RD becomes strong enough to make an
initially subcritical cloud clump supercritical and induce its collapse. Our
results indicate that the formation of a supercritical core is regulated by a
complex interplay between gravity, self-gravity, the magnetic field strength
and nearly transonic and trans-Alfv\'enic turbulence, confirming that RD is
able to remove magnetic flux from collapsing clumps, but only a few of them
become nearly critical or supercritical, sub-Alfv\'enic cores, which is
consistent with the observations. Besides, we have found that the supercritical
cores built up in our simulations develop a predominantly helical magnetic
field geometry which is also consistent with observations. Finally, we have
evaluated the effective values of the turbulent reconnection diffusion
coefficient and found that they are much larger than the numerical diffusion,
especially for initially trans-Alfv\'enic clouds, ensuring that the detected
magnetic flux removal is due to to the action of the RD rather than to
numerical diffusivity.
