Was changing the oil today on my van today ( which I failed to do due to a previous owner cross threading then rounding off te sump plug) And i noticed
this pipe

was broken as you can see and looked it was attached to here

and the air intake.
My question is what is this pipe for ? What is it called so I can buy a new one ? And finally if it had been broken for a while explain why my engine is running like a bag of tripe. When I put my hand over the exhaust manifiold it is sucking in air so is it some sort of vacuum system?
You responses are Much appreciated - please be gentle I am not very knowledgeable when it comes to engines
Its a 1.9 DG engine