Total variation (TV)-based CT image reconstruction, employing the image gradient sparsity, has shown to be experimentally capable of reducing the X-ray sampling rate and removing the unwanted artifacts, yet may cause unfavorable over-smoothing and staircase effects by the piecewise constant assumption. In this paper, we present a total generalized p-variation (TGpV) regularization model to adaptively preserve the edge information while avoiding the staircase effect. The new model is solved by splitting variables with an efficient alternating minimization scheme. With the utilization of generalized p-shrinkage mappings and partial Fourier transform, all the subproblems have closed solutions. The proposed method shows excellent properties of edge preserving as well as the smoothness features by the consideration of high order derivatives. Experimental results indicate that the proposed method could avoid the mentioned effects and reconstruct more accurately than both the TV and TGV minimization algorithms when applied to a few-view problem.