On a divisibility relation for Lucas sequences

In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where $\mathbf{U}:=\{U_n{\}}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.