  A spin-1/2 Heisenberg model on honeycomb lattice is investigated by doing
triplon analysis and quantum Monte Carlo calculations. This model, inspired by
Cu$_2$(pymca)$_3$(ClO$_4$), has three different antiferromagnetic exchange
interactions ($J_A$, $J_B$, $J_C$) on three different sets of nearest-neighbour
bonds which form a kagome superlattice. Interestingly, while the model is
bipartite and unfrustrated, its quantum phase diagram is found to be dominated
by a quantum paramagnetic phase that is best described as a spin-gapped
hexagonal-singlet state. The N\'eel antiferromagnetic order survives only in a
small region around $J_A=J_B=J_C$. This study provides an understanding of the
gapped non-magnetic behaviour observed in Cu$_2$(pymca)$_3$(ClO$_4$) at low
temperatures.
