Sun's Effect on Water Vapor in the Air
You need to define what you mean by 'adding more sun'. I'm asuming you mean 'increase the solar constant'.

The amount of water vapour in the air is, as a first order approximation, related to temperature - every degree of temperature rise gives us 7% more water vapour.  So if 'adding more sun' increases the temperature then it automatically increases water vapour. 

Now, to add more sun without increasing the temperature, you'd have to compensate by doing something like painting large areas of land white; this would increase the Earth's albedo, reflecting the extra solar radiation. No temperature change overall, so no water vapor change.

The only direct effect of the sun on water vapor is something entirely different - through photo-dissociation. This is where the sun's UV rays directly break up water molecules.  However, in order to have an appreciable effect on the day to day concentration of water vapour, you'd need to replace the Sun with a much bigger and hotter star that blasted out far more UV radiation. The problem here is that everyone would die.

So as far as the question can be answered, it would be no under any realistic scenario.