We report the recent progress in relativistic mean-field (RMF) and beyond approaches for the low-energy structure of deformed hypernuclei. We show that the Λ hyperon with orbital angular momentum ℓ = 0 (or ℓ > 1) generally reduces (enhances) nuclear quadrupole collectivity. The beyond mean-field studies of hypernuclear low-lying states demonstrate that there is generally a large configuration mixing between the two components [A-1Z(I+) ⊗ - Λp1/2]J and [A-1Z(I ± 2+) ⊗ - Λp3/2]J in the hypernuclear 1/21-, 3/21- states. The mixing weight increases as the collective correlation of nuclear core becomes stronger. Finally, we show how the energies of hypernuclear low-lying states are sensitive to parameters in the effective NΛ interaction, the uncertainty of which has a large impact on the predicted maximum mass of neutron stars.