That is your purge valve. There is a pipe which runs to a charcoal canister under the passenger side wing (the pipe that you have taped up). This canister receives fuel vapour from the petrol tank vent, and then every so often under certain driving conditions, the ECU will operate the purge valve. The other pipe from the purge valve runs along the back of the rocker and into the inlet, so that the purged vapour is burned by the engine.
They commonly break at that point, mine was broken too. I discovered however that the broken part wasn't really required in that I could remove the broken piece of plastic and then just fit the hose on directly with a pipe clip. I can't remember exactly how that was the case, but I'm sure if you unbolt the purge valve so you can work with it off the car, you'll see what I mean.
The effect of this pipe being broken would be that when the valve is opened by the ECU, it could create a sort of vacuum leak, however this would come and go as the ECU modulated the purge valve, and I don't know if it would be a significant vacuum leak or not (depends of the path into the inlet is restricted or not). I seem to remember that the ECU modulates the purge valve under cruising conditions, not accelerating conditions, but I'm not sure.
If the valve is not working, or the electrical connection to it is broken, then your broken pipe would have no effect at all on the car's running, as it would never open, and the charcoal canister would just vent to the air through the broken pipe.
The fact that you have found this though, is a warning to you to check all the breather hoses as they are a common culprit for splitting, breaking etc and could all in theory give you issues.