That vacuum hose links a solenoid valve (the thing with two wires) to a vacuum diaphragm (in the big horizontal round black casing) which moves a flap-valve beneath it, in the Secondary Air Injection (SAI) system.
What should happen is that the ECU tells that solenoid valve to open and the SAI pump to start up when the engine is started from cold. Once that solenoid valve opens, it allows a manifold vacuum signal to be passed to the diaphragm which opens the flap valve. This allows the pumped air into the exhaust manifold via a non-return valve (the big round silver thing). This reduces cold-start emissions by warming the cat rather rapidly.
After a short time (a minute?) the ECU shuts down the pump and allows this solenoid to close, shutting the flap valve. The flap valve (I think) is to prevent air being sucked into the exhaust system by pressure pulses during normal driving, which would interfere with lambda sensing and closed loop fuelling.
I can't remember where the hinge of the flap valve is, but it may be that if that flap valve is not opening - because of the broken vacuuum connection - then the SAI pump may be pumping into a 'dead end' which might cause it to draw excessive current and make bad noises.
Having said all that, without hearing the 'cranking noises' you describe, this fault may be totally unrelated.
The fact that you are seeing the engine management light on means your ECU knows of a fault, and can tell you what it thinks is not working if you have a paperclip. see maintenance guides index for "Paperclip Test".
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