This paper aims to model the degradation paths of degrading units in presence of an unexplained form of heterogeneity among the paths. Focus is on monotonic increasing degradation processes where the degradation increments over disjoint time intervals are not independent. The degradation path of each unit is described via the Transformed Gamma process, and the "age" and "state" functions that characterize the Transformed Gamma process are here assumed to be power-law functions. The unexplained heterogeneity among paths of different units is accounted for assuming that the scale parameters of the "age" and "state" functions vary from unit to unit. This variability is modeled assuming that the scale parameters are independent gamma random variables. Under these assumptions, a quite mathematically tractable model is obtained. The main properties of the proposed model are discussed, and inferential procedures based on the maximum likelihood criterion are presented. Finally, the proposed model is applied to a real set of degradation data to show the feasibility of the proposed model.