We have carried out an extensive X-ray spectral analysis of a sample of galaxies exhibiting molecular outflows (MOX sample), to characterize the X-ray properties and investigate the effect of AGN on the dynamical properties of the molecular outflows. We find that the X-ray bolometric correction (L2-10keV/L-AGN) of these sources ranges from similar to 10(-4.5) to 10(-0.5), with similar to 70% of the sources below 10(-2), implying a weak X-ray emission relative to the AGN bolometric luminosity (L-AGN). However, the upper limit on the 2-10keV luminosity (L2-10 (keV), (12) (mu m)) obtained from 12 mu m flux, following the correlation derived by Asmus et al., are similar to 0.5-3 orders of magnitude larger than the L2-10 (keV) values estimated using X-ray spectroscopy, implying a possibility that the MOX sources host normal AGN (not X-ray weak), and their X-ray spectra are extremely obscured. We find that both L2-10 (keV), and L-AGN correlates strongly with the molecular outflow velocity as well as the mass outflow rates (M-out), implying that the central AGN plays an important role in driving these massive outflows. However, we also find statistically significant positive correlations between the starburst emission and MO mass outflow rate, L-Starburst vs M-out, and L0.6-2keV vs M-out, which implies that starbursts can generate and drive the molecular outflows. The correlations of MO velocity and M-out with AGN luminosities are found to be stronger compared to those with the starburst luminosities. We conclude that both starbursts and AGN play crucial role in driving the large-scale MO.