The primary purpose of a secondary air injection system is to consume the excess fuel that is injected during cold-start enrichment before it can reach the catalytic converter. This has the effect of improving tailpipe emissions.
There are some undesired consequences of excess fuel reaching the cats:
- high concentration of unburnt hydrocarbons (HC)
- high concentration of carbon monoxide (CO)
- fouling of the catalytic converter reaction surface, which reduces its efficacy
Injecting air into the exhaust manifold tends to react with some of the excess HC, CO present in the exhaust gas. It does not burn off all of it, but does improve emissions. Engine bay/exhaust manifold temperatures increase as a result of this reaction.
Regarding fuel-injected engines...
Bosch claims that its fuel injection systems do not require secondary air injection for closed-loop operation unless engine displacement is relatively large¹. This is because the air-fuel ratio metering is fine enough for excess, unburnt hydrocarbons to not be an issue.
I can tell you that the 5.0 L BMW S62 is an example of a Bosch-managed engine that comes equipped with secondary air injection, but it is only active when running in open-loop (e.g. cold-starts).