We numerically analyze two-dimensional photonic crystal (PhC) nanocavities on glass with a thin glass film on top of the structure. We investigated a multistep heterostructure GaAs PhC nanocavity located on glass. We found that covering the structure even with a very thin glass film efficiently suppresses unwanted polarization mode conversion occurring due to the asymmetric refractive index environment around the PhC. We also uncovered that the glass-covered structure can exhibit a higher Q factor than that observed in the structure symmetrically cladded with thick glass. We point out that the mode mismatch between the PhC nanocavity and modes in the upper glass film largely contributed to the observed Q-factor enhancement. These observations were further analyzed through the comparison among different types of on-glass PhC nanocavities covered with thin glass films. We also discuss that the in-plane structure of the upper glass film is important for additionally enhancing the Q factor of the nanocavity.