  The $^{120}$Sn($p$,$p\alpha$)$^{116}$Cd reaction at 392 MeV is investigated
with the distorted wave impulse approximation (DWIA) framework. We show that
this reaction is very peripheral mainly because of the strong absorption of
$\alpha$ by the reaction residue $^{116}$Cd, and the $\alpha$-clustering on the
nuclear surface can be probed clearly. We investigate also the validity of the
so-called factorization approximation that has frequently been used so far. It
is shown that the kinematics of $\alpha$ in the nuclear interior region is
significantly affected by the distortion of $^{116}$Cd, but it has no effect on
the reaction observables because of the strong absorption in that region.
