Sounds like part of the Secondary Air Injection system.
During cold start an air pump under the nearside front corner of the car sends air into the exhaust manifold to create an 'afterburner' effect which helps heat up lambda sensor and (mainly) catalytic converter more rapidly than it would otherwise, so reducing hydrocarbon emissions which can be considerable at this operating phase.
There are two valves in the system, a non-return valve to stop exhaust fumes coming out towards the rubber pipework (the metal round bit you mention), and a flap valve at the other end of your dodgy hose which is controlled by manifold vacuum mediated by a solenoid valve. This latter valve is oriented and controlled so that air cannot pass into the exhaust except during the minute or so of SAI operation at coldstart, as this would otherwise confuse the lambda sensor and stuff up your mixture control.
Not sure why your hose appears to have collapsed, maybe just degradation of the rubber through heatsoak?