  We develop a variational approach at finite temperature that incorporates
many-body correlation self-consistently. The grand potential is constructed in
terms of Green's function expressed by the variational parameters. We apply
this formalism to weakly interacting Bose-Einstein condensates to incorporate
the dynamical 3/2-body processes, which are considered important in the
dynamical properties. The processes lower the free energy below the mean-field
Hartree--Fock--Bogoliubov's value in the same way as a previous
zero-temperature formalism. From our numerical results, the pair creation or
annihilation processes neglected in the Popov--Shohno approximation are
enhanced, particularly in the long wavelength region, owing to the many-body
effects. Because the 3/2-body correlations give a finite contribution to the
self-energy of quasiparticles, they may change the microscopic properties
qualitatively, even in the weak-coupling region.
