Summary: In this paper, we present sufficient conditions for the oscillation and asymptotic behaviour of solutions of delay dynamic equations of the form \[ [x(t)+A(t)x{\alpha}(t))]^{\Delta^n}+B(t)x{\beta}(t))=0 \quad {\text{for }}t\in [t_0,\infty)_{\mathbb T}, \] where \(n\in \mathbb N, \mathbb T\) is a time scale unbounded above, \(t_0\in \mathbb T,A\in \mathrm{C}_{\mathrm{rd}}([t_0,\infty)_{\mathbb T}, \mathbb R)\) satisfies either \(A\in \mathrm{C}_{\mathrm{rd}}([t_0,\infty)_{\mathbb T},[0,1]_{\mathbb R})\) with \(\limsup _{t\to \infty} A(t)<1\) or \(A\in \mathrm{C}_{\mathrm{rd}}([t_0,\infty)_{\mathbb T},[-1,0]_{\mathbb R})\) with \(\liminf_{t \to \infty} A(t)>-1, B\in \mathrm{C}_{\mathrm{rd}}([t_0,\infty)_{\mathbb T}, \mathbb R_0^+)\) and \({\alpha},{\beta}\in \mathrm{C}_{\mathrm{rd}}([t_0,\infty)_{\mathbb T}, \mathbb T)\) are unbounded nondecreasing functions such that \({\alpha}(t),{\beta}(t)\leqslant t\) for all \(t\in [t_0,\infty)_{\mathbb T}\) under the condition \[ \int _{t_0}^\infty B({\eta}){\Delta}{\eta}=\infty . \]