We have this vehicle with the below fault codes, but I am unsure what they relate to

P0810 malfunction of secondary air injection at right bank (function chain)
P0814 malfunction of secondary air injection at left bank (function chain)

This car has already had a secondary air injection pump and relay fitted by another garage and the main agents have performed a PCM software update.

We have done provisional checks, smoke test, plugs, air filter ect.

Before I get too deep into someone else's problem I would like to get some clarity on the logged fault codes.

Regards

Simon