We explore the effect of poly (ethylene glycol) (PEG) molar mass on the intrinsic permeability and structural characteristics of poly (ethylene glycol) diacrylate PEGDA/PEG composite hydrogel membranes. We observe that by varying the PEG content and molar mass, we can finely adjust the water intrinsic permeability over several orders of magnitude. Notably, we show the existence of a maximum water intrinsic permeability, already identified in a previous study to be located at the critical overlap concentration C^* of PEG chains, for the highest PEG molar mass studied. Furthermore, we note that the maximum intrinsic permeability follows a non-monotonic evolution with respect to the PEG molar mass and reaches its peak at 35 000 g.mol-1. Besides our results show that a significant fraction of PEG chains is irreversibly trapped within the PEGDA matrix even for the shortest molar masses down to 600 g.mol-1. This observation suggests the possibility of covalent grafting of PEG chains to the PEGDA matrix. CryoSEM and AFM measurements demonstrate the presence of large micron-sized cavities separated by PEGDA-rich walls whose nanometric structure strongly depends on the PEG content. By combining our permeability and structural measurements, we suggest that the PEG chains trapped inside the PEGDA rich walls induce nanoscale defects in the cross linking density, resulting in an increased permeability below C^*. Conversely, above C^*, we speculate that partially-trapped PEG chains may form a brush-like arrangement on the surface of the PEGDA-rich walls, leading to a reduction in permeability. These two opposing effects are anticipated to exhibit molar-mass-dependent trends, contributing to the non-monotonic variation of the maximum intrinsic permeability at C^*. Overall, our results demonstrate the potential to fine-tune the properties of hydrogel membranes, offering new opportunities in separation applications.