Distilled water under our atmosphere usually has a pH around 5.6 or so because carbon dioxide reacts with water to form carbonic acid, which dissociates to lower the pH. In carbonated water, the concentration of carbonic acid is higher because there's more carbon dioxide around, hence the lower pH. If you adjust the pH by adding alkaline, you will shift the equilibrium toward carbon dioxide. Basically, you will decarbonate the water. You will still have carbonic acid but nearly as much. pH is a log scale so adjusting from pH 4.2 to pH 6.0-6.5 is roughly a 100fold less acidic. So I guess it depends on the concentration of carbonic acid you want to deliver to the plant. Of course, you will still have plant biology and chemistry to contend with, so having more or less carbonic acid in your water may or may not affect how much carbonic acid makes it to wherever it needs to go.
Just out of curiosity, what makes you think carbonic acid in your water will "increase photosynthesis"?