The authors consider the following nonlinear neutral delay difference equation \[ \Delta(a_n(\Delta(x_n+ p_nx_{n-\tau}))^\gamma)+ q_n x^\gamma_{n-\sigma}= 0.\tag{1} \] By means of Riccati transformation techniques they establish discrete Kamenev-type oscillation criteria for (1) in the linear and superlinear case \((\gamma\geq 1)\). Further, they consider the slightly more general equation \[ \Delta(a_n(\Delta(x_n+ p_n x_{n-\tau}))^\beta)+ q_n x^\gamma_{n-\sigma}= 0.\tag{2} \] They obtain sufficient conditions for the oscillation of all solutions of (2) in the case \(0<\gamma< 1\) and \(\beta\geq 1\). The results generalize and improve many of the known oscillation criteria.