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The correct operation and high performance of complex superconducting integrated circuits significantly depend on fabrication-process-induced variations of the Josephson junction critical current Ic. Such variations in Nb/Al/AlOx/Nb junctions were investigated and shown to be dependent on how the junction electrodes are connected to other layers in the integrated circuit, especially the ground plane and the Ti/Pd/Au contact pad. The observed enhancement of Ic and gap voltage over time for junctions with certain wiring connections suggests that the phenomenon is related to the diffusion over time of impurities between the junction electrode and the Ti/Pd/Au pad. Considering the strong affinity of both Nb and Ti to H, a model where H is the main impurity element involved in the diffusion-related phenomenon is presented. The results show that direct wiring to Ti is sufficient to observe Ic variations. The results also suggest that as fabricated, the interface between the junction counter-electrode and the AlOx barrier is already close to or at full H saturation, significantly depressing the Ic by up to 20%, compared to clean Nb junctions.