The stoichiometrics of respiration say that the CO2 exhaled by one person, plus their poop, can be converted by algae into 3000 calories of nutrition and 26 moles of oxygen (enough for a day) using 220kWh per day. See Algae Bioreactor for O2 and CO2 (and food?) - Any data on required size and power requirements?
The answer to your question is that a person exhales enough CO2 for the algae system to potentially regenerate all the O2 they require without adding additional CO2.
An algae bioreactor initially seems like a great idea for space travel. It is very elegant to turn waste CO2 and astronaut poop into oxygen and food. Unfortunately, the device’s mass and energy requirements make it impractical for voyages in the inner solar system.
On Mars, the need to produce industrial quantities of oxygen would make photosynthetic oxygen unnecessary. Matt Damon's poop would still be needed for growing potatoes.