Plasma focus experiments were carried out at a modified PF-1000 where the cathode disc was added in front of the anode. Experimental results indicated a fraction of thermonuclear neutrons on the mega-ampere current level. In order to prove the thermonuclear mechanism, the time of neutron production and the neutron energy spectrum were measured by time-of-flight (TOF) diagnostics. Neutron TOF signals showed that the neutron production was a multiphase process and more than one mechanism occurred simultaneously. The occurrence of the thermonuclear mechanism was most evident during the plasma stagnation at low deuterium pressures. At low filling pressures, the narrow width of the neutron energy spectra demonstrated an ion temperature of about 1 keV. The possibility of thermonuclear neutrons was studied also after the stagnation, during the main neutron emission. In this case, the thermonuclear mechanism could be verified by calculating the number of deuterons that participate in the fusion process. For the bulk of thermonuclear plasmas, a significant fraction of plasma should participate in fusion. Finally, the basic consideration of the thermonuclear mechanism in Z-pinches showed the reasonableness of the MagLIF concept.