If an engine has to run extra rich, an air pump adds enough air to balance the air/fuel ratio. If the car uses a cat, the extra heat generated by the process shortens catalyst light off time. Some cars simply need them operating continuously because they don't burn fuel well at all, but that typically applied to 1960s designed engines that the manufacturer wanted to keep producing. The last of the Ford SBF, BBF, and I6 are the first examples I can think of.
It seems every modern German car I have worked on has an electric air pump, even ones with cutting edge technology.