  We investigate the interplay between strong electron correlations and
charge-lattice interaction in cuprates. The coupling between half breathing
bond stretching phonons and doped holes in the t-t'-J model is studied by
limited phonon basis exact diagonalization method. Nonadiabatic electron-phonon
interaction leads to the splitting of the phonon spectral function at half-way
to the zone boundary at $\vec{q}_s=\{(\pm \pi / 2, 0), (0, \pm \pi / 2) \}$ and
to low energy kink feature in the electron dispersion, in agreement with
experimental observations. Another kink due to strong electron correlation
effects is observed at higher energy, depending on the strength of the
charge-lattice coupling.
