I believe it's called the bypass hose, it supplies coolant to the manifold to keep it cool, since it's a big block of aluminium (usually) it needs cooling as it gets heat soak from the engine itself.
Hot inlet= low pressure on the compression stroke=lowered power from the burn.
Hence intercoolers on turbo cars.
Some of the other Mazda's use water for idle control and as a auto choke on cold mornings.