  It remains an open question whether the two-dimensional single-band pure
Hubbard model and its related pure $t$-$J$ model truly capture the
superconducting order in cuprates. Recent numerical studies on this issue have
raised a notable disparity in superconducting order between the pure Hubbard
model and the pure $t$-$J$ model. Inspired by these, we investigate the role of
the three-site hopping term in $d$-wave superconductivity, such a term is
usually neglected in the effective Hamiltonian of the Hubbard model, though its
amplitude is of the same order as the superexchange coupling $J$ in the $t$-$J$
model. Our slave-boson mean-field solution demonstrates the suppression of
$d$-wave superconducting order by incorporating the three-site hopping term,
consistent with numerical observations by the density matrix renormalization
group. This suppression could be understood as a result of competition between
superexchange interaction and three-site hopping, the former favors $d$-wave
pairing while the latter favors $s$-wave pairing. We also discussed its role in
quasiparticle dispersion and boson-condensation temperature. Our findings may
offer an alternative understanding of the recent numerical contrasting findings
in the strong coupling regime: the absent or weak superconductivity in the pure
Hubbard model, while the robust superconductivity in the $t$-$J$ model without
including the three-site hopping term.
