"Actually , after doing a little homework, the pump oxidizes the HCs in the catalysts and helps them heat up faster from a cold start."
This description is absolutely acurate, though maybe incomplete.
For years smog pumps(A.I.R. Pumps - air injection reaction) were used continuously. They introduced air (oxygen) to the partially burned products of combustion CO and HC. The heat of the exhaust promoted a certain amount of further oxidation to CO2 and H2O. This was inefficient and catalysts were introduced to help.
This worked well BUT the catalyst and oxygen only made the creation of oxides of Nitrogen worse. Thus came the three way catalyst and the lambda oxygen control system.
The lambda system can not have extra O2 added as this is the measurement to establish stoiciometry: the exact amount of fuel and air to creat the exact amount of HC and CO2 the accepted by-product. During closed loop lambda control the NOX (oxides of nitrogen) are reduced (the chemical oposite of oxidize)by catalytic action to there base elements N2 and O2. This O2 and a small extra amount of O2 (that is fed during the lean switchoever phase of closed loop) feeds the oxidation portion of the cat to finish the combustion of unburned HC and CO>
The current action of the A.I.R Pump (or currently called: secondary air pump)is as described. It acts as the first simple oxidation action (before the cat is hot and before the O2 sensor is capable (hot). It does a sufficient job to reduce the HC content during the dirtyest performance time.
Most important it causes the catalyst to heat more quickly by creating this fire inside it. Hot air plus fuel plus slight cat activity equals great heat source.