name the part and function please

[Braincrank]

I would like some info on these two parts please. First, what does the black tube do (the one with the two arrows)?
I noticed that it is broken where I drew the red arrow. It comes from the air filter.
Second, there's a little tube (looks like a vacuum pipe) coming from where I drew the green arrow, and it goes into the little round barrel/cylinder that you can see in the second picture. What does that little barrel/cylinder do?
A couple of months ago the car didn't start very well and after searching long and hard I found the connection from that tiny tube that comes out of that little barrel and goes to the black tube (2. pic , green arrow) to be broken. After fixing it the car was starting ok again.
I was a bit baffled but nevertheless happy.
Does the tube in pic 2 or the little tube that goes into pic 1 influence the starting or running of the engine ?
I ordered a new tube today and I should have it by Wednesday (-:
Will it be ok to drive the car or can I get stuck because of that broken tube ?
Any info is welcome (-:

 

[martyz]

The small tube is a vacuum pipe to the fuel pressure regulator.
The large black pipe is from the airbox to engine vent to aux air device.
both will cause poor starting/rough uneven idle,cutting out.There is another pipe
from the aux air device to the airflow meter(underneath the unit) that can cause the same fault as well.