What does the pollution pump on my 1969 Nova do?

It's abandoned in place, but is there any advantage to having it hooked up?

It pumps metered air into the exhaust manifolds to aid in emmision control through secondary oxidation of unspent hydro carbons. If you do not live in the crazy emissions states like NY or CA, it's useless and annoying. be sure you cap the exhaust ports with brass fittings when you take it off and put it up in the garage rafters for afterward collector use. certainly save all the parts for future restoration.

ASE Cert Auto Tech

It keeps a belt tight against the water pump for safety, and it's suppossed to injecr new air in the exhaust system to clean up the air

Lots of people used to disconnect them. Non mechanics although all the anti pollution stuff was bad for the engine. Never mind that pollution was bad for their lungs.
The pump circulates some of the exhaust back thru the intake to better burn off pollutants and unburned gas.
Your engine was intended to run with it, and it does run better with it connected.
In some states and cities, it must be hooked up to pass inspection.
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To connect it:
First be sure it is not seized up, which can be one reason why it is disconnected.
Make sure the hoses are in good form, if they leak it is useless and will make the engine run bad.
Check the one way valve, it is the metal part that projects external and is held on by screws. be sure it is not clogged with carbon, if it's, this will also make the engine run bad.
Make sure you rout the belt properly and it runs straight and true.
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It will make your engine run hotter with it.

sci has the correct answer. They pumped cold air into the exhaust manifolds supposedly to decrease emissions. In reality, they did nothing worth putting them on for. Most locked up and became like yours, just extra weight. Some went bad and made poppeling noises in the exhaust pipes. Quite irritating and costly to replace. Most just cut the belts and forgot about them. The belt typically only drove the pump and nothing else. It was a squib to make the govt. think they doed something about emissions, like Chrysler's balky "electronic air cleaner boxes". Still today, cow farts do more to hurt the air than cars.