Can you put plants that grow under water into the system to oxygenate water as well as give smaller fish such as mosquito fish an area to live and hatch?

The only problems I can think of would be the amount of light those plants would need to produce enough O2 for the fish and the possible clogging factor of the water carrying pipes.

The photosynthesis would require enough light and that might lead to overheating of the water, which would be fatal to the fish and possibly the plants in the system.
As the O2 producing plants grew they could get into the pipes of the system and consequently clog those pipes enough as to reduce the flow enough to be an issue.

Redhawk

Ever wonder why Cattails are so successful in wetlands and hypoxic soils,  they have Aerenchyma and get enough oxygen into the rhizophere to make them successful.   There may be other plants like that but it is the only one I know of.

It is CO2 that we need for plants to grow underwater there are several youtube videos that address this for plants in aquariums.  These are energy intensive and also require more time and attention than i could justify for a balanced self organizing system however they may provide clues. Remember photosynthesis gives off Oxegen.   

hau Jim. You do realize that Photosynthesis is only half of plant respiration yes?  In the presence of Sun light (the complete spectrum isn't needed since plants are more photo active in the red range). Plants will give off O2, in the dark they use O2 and give off CO2, thus completing a cycle.
An aquarium is quite different than an aquaponics system since it is designed to grow plants for food and uses fish for the fertilizer for those plants.

Not sure how it effects the O2 levels, but bacopa grows well in my aquarium with low light levels.  It doesn't grow as much without sun, but it does grow.

Also, the little fishies don't nibble on it because the leaves are extra fleshy.