The question shows confusion about the purpose of the splitting of water in photosynthesis. Quoting from Berg et al., in the light reactions of photosynthesis
“Light is absorbed and the energy is used to drive electrons from
water to generate NADPH and to drive protons across a membrane. These
protons return through ATP synthase to make ATP.”
The useful product of splitting water is energy.
There is no problem in the provision of hydrogen ions for biochemical reactions as such — they are available from the dissociation of water.
The diagram above, summarizing the light reactions may be helpful, although it conceals detail and introduces the complexity of energy generation. It shows the (thalkaloid) membrane separating the location of the reduction of NADP+ (outside) from where water is split (inside).
The important product of the reaction of light photons with water is excited electrons. The chemical products are relatively unimportant — certainly for oxygen, and the only importance of the hydrogen ions produced is their location within the thalkaloid, lowering the pH.
The initial fate of the hydrogen atom is complex, and there a many intermediates to which the excited electron is transferred before, ultimately to NADP+:
NADP+ + H+ + 2e– → NADPH
It is important to realize that the energy is used to reduce NAD to NADPH, that NADPH can be regarded as a special type of ‘high-energy compound’ — not a general one like ATP, but one that can specifically drive the reduction of carbon backbones needed in synthesis.
The misapprehension is that there the hydrogen ions for the above reaction are those produced in the splitting of water, or that there is in general any problem in obtaining hydrogen ions for biochemical reactions. Biology exists in an aqueous environment, and water dissociates into hydrogen and hydroxyl ions.
H2O ⇌ H+ + OH–
So hydrogen ions are irrelevant to the reduction reactions involved in the conversion of carbon dioxide to carbohydrates in the light-independent reactions of photosynthesis (they are actually produced in NADPH reductions.). It is NADPH that is supplying the hydrogen component of the carbohydrates.
The diagram also brings in a complexity that needs a book chapter to explain: the hydrogen ion gradient generated between the inside and outside of the thalkaloid space provides the energy to phosphorylate ADP to ATP. The ATP is relevant to the light-independent reactions of photosynthesis in that it ultimately provides the energy for formation of the carbon–carbon bonds.