  The direct reaction theory widely used to study single-particle spectroscopic
strength in nucleon transfer experiments is based on a Hamiltonian with
two-nucleon interactions only. We point out that in reactions with a
loosely-bound projectile, where clustering and breakup effects are important,
an additional three-body force arises due to three-nucleon ($3N$) interaction
between two nucleons belonging to different clusters in the projectile and a
target nucleon. We study the effects of this force on nucleon transfer in
$(d,p)$ and $(d,n)$ reactions on $^{56}$Ni, $^{48}$Ca, $^{26m}$Al and $^{24}$O
targets at deuteron incident energies between 4 and 40 MeV/nucleon. Deuteron
breakup is treated exactly within a continuum discretized coupled-channel
approach. It was found that an additional three-body force can noticeably alter
the angular distributions at forward angles, with consequences for
spectroscopic factors' studies. Additional study of transfer to $2p$ continuum
in the $^{25}$F$(p,2p)^{24}$O reaction, involving the same overlap function as
in the $^{24}$O($d,n)^{25}$F case, revealed that $3N$ force affects the $(d,n)$
and $(p,2p)$ reactions in a similar way, increasing the cross sections and
decreasing spectroscopic factors, although its influence at the main peak of
$(p,2p)$ is weaker. The angle-integrated cross sections are found to be less
sensitive to the 3N force contribution, they increase by less than 20$\%$.
Including 3N interactions in nucleon removal reactions makes an essential step
towards bringing together nuclear structure theory, where 3N force is routinely
used, and nuclear direct reaction theory, based on two-nucleon interactions
only.
