We investigate the properties of accretion flows on to a black hole (BH) with a mass of M$_BH$ embedded in an initially uniform gas cloud with a density of n$_\ensuremathınfty$ in order to study rapid growth of BHs in the early Universe. In previous work, the conditions required for super-Eddington accretion from outside the Bondi radius were studied by assuming that radiation produced at the vicinity of the central BH has a single power- law spectrum \ensuremathν$^-\ensuremathα$ at h\ensuremathν \ensuremath≥ 13.6 eV (\ensuremathα ̃ 1.5). However, radiation spectra surely depend on the BH mass and accretion rate, and determine the efficiency of radiative feedback. Here, we perform two- dimensional multifrequency radiation hydrodynamical simulations taking into account more realistic radiation spectra associated with the properties of nuclear accretion discs. We find that the critical density of gas surrounding the BH, above which transitions to super-Eddington accretion occur, is alleviated for a wide range of masses of seed BHs (10 \ensuremathłesssim M$_BH$/M$_☉$ \ensuremathłesssim 10$^6$) because photoionization for accretion disc spectra are less efficient than those for single power-law spectra with 1 \ensuremathłesssim \ensuremathα \ensuremathłesssim 3. For disc spectra, the transition to super-Eddington is more likely to occur for lower BH masses because the radiation spectra become too hard to ionize the gas. Even when accretion flows are exposed to anisotropic radiation, the effect due to radiation spectra shrinks the ionized region and likely leads to the transition to a wholly neutral accretion phase. Finally, by generalizing our simulation results, we construct a new analytical criterion required for super- Eddington accretion; (M_BH/105̂ M_☉) (n_\\ensuremathınfty\/10\ ̂cm\^3̂\) \ensuremath\gtrsim 2.4 (< \ensuremathın > /100 eV)\^\/̂9\, where <\ensuremathın> is the mean energy of ionizing radiation from the central BH.