We study the static and dynamic properties of a kink in a chain of harmonically coupled atoms subjected to a deformable double-well substrate potential. We treat intrinsically the lattice discreteness without approximation and show that in some deformation-parameter ranges each period of the PN (Peierls-Nabarro) potential consists of two wells whose minima are located respectively on a lattice site and midway between two adjacent sites of the chain. In some other parameter ranges each period of the PN potential posseses a single well whose minimum is located either on a lattice site or midway between two adjacent lattice sites. We examine the kink's dynamics by using a multiple-collective-variable treatment, that is, we derive the exact equations of motion for the collective variables X and Y - which describe respectively the center-of-mass mode and the internal mode of the kink. We numerically solve the collective variable equations of motion for the trapped and untrapped regimes of the discrete-kink motion, and show that the presence of a nonlinear internal mode makes a contribution of particular importance in the discrete-kink's dynamics. Indeed, we show that during its untrapped regime, the discrete kink can undergo one or more temporary trappings and even a reflection back over several PN wells, and relate such behaviours to the effects of the excitations of the internal mode of the kink.