What is the hose, that connects to the back of the air intake, before the maf?
Is it needed for correct operation?

 

Thats for your A.I.R. system, no its not "needed" nor is the entire AIR system.

 

If it's what I think you refer to, it's the AIR pump hose. It supplies filtered air to your AIR pump, which is an emissions control device. The hose is needed for this device to work properly. Many people remove the whole pump assembly and plug the holes if they aren't worried about emissions.