We explore how to configure an argon atmospheric-pressure plasma jet for enhancing its production of hydrogen peroxide (H2O2) in deionised water (DIW). The plasma jet consists of a quartz tube of 1.5 mm inner diameter and 3 mm outer diameter, with an upstream internal needle electrode (within the tube) and a downstream external cylindrical electrode (surrounding the tube). The plasma is operated by purging argon through the glass tube and applying a sinusoidal AC voltage to the internal needle electrode at 10 kV (peak-peak) with a frequency of 23.5 kHz. We study how the following operational parameters influence the production rate of H2O2 in water: tube length, inter-electrode separation distance, distance of the ground electrode from the tube orifice, distance between tube orifice and the DIW, argon flow rate and treatment time. By examining the electrical and optical properties of the plasma jet, we determine how the above operational parameters influence the major plasma processes that promote H2O2 generation through electron-induced dissociation reactions and UV photolysis within the plasma core and in the plasma afterglow; but with a caveat being that these processes are highly dependent on the water vapour content from the argon gas supply and ambient environment. We then demonstrate how the synergistic action between H2O2 and other plasma generated molecules at a plasma induced low pH in the DIW is highly effective at decontaminating common wound pathogens Gram-positive Staphylococus aureus and Gram-negative Pseudomonas aeruginosa. The information presented in this study is relevant in the design of medical plasma devices where production of plasma reactive species such as H2O2 at physiologically useful concentrations is needed to help realise the full clinical potential of the technology.