Using random effects in the modeling of homogeneous Poisson processes (HPP) has proved effective (Cook and Lawless (2007) and Gongjun et al. (2015)). We (Khribi et al. (2015)) compared the truncated normal prior (truncated at 0) to the usual gamma prior in the prediction for homogeneous Poisson processes and we concluded that the truncated normal prior (which is equivalent to the 2-moment maximum entropy prior) compared very favorably to the gamma one. This method was called the general Poisson-MaxEnt model. Unfortunately, because of the 2-moment condition on our maximum entropy prior, we were restricted to considering only cases where the coefficient of variation was less than or equal to 1 (Wragg and Dowson (1970)). Here we remove this restriction by the use of the \(k\)-moment maximum entropy prior (\(k>2\)). The effectiveness of the general Poisson-MaxEnt model with this \(k\)-moment prior for prediction in HPP was measured by two goodness-of-fit criteria: Kullback-Leibler divergence and a discrepancy measure. The estimators obtained by these methods are compared to the estimators obtained with the two moment maximum entropy prior and the gamma prior used by (Khribi et al. (2015)). The likelihood ratio test is used in order to determine when to stop adding higher order moments. We also illustrated on two examples: one concerning the occurrence of mammary tumors in laboratory animals taking part in a carcinogenicity experiment and the other, a warranty data set from the automobile industry.