guys, what is the hose in the pic below for?

for mine the one connected to engine (right beside the engine oil cap) was loose.... i tried to push it in but it was freaking difficult....

was it supposed to be loose? i dun think so....


Attached image(s)

can elaborate further??

coz when i take a look at the connector, there's some engine oil there....
i saw some ppl install air-filter-like gadget there... is that supposed to help in terms of engine response?

thanks for the info guys......
at last it cleared up my confusion

i tried driving the car with the connector loose (but not letting the gaphole exposed totally)... didnt have much difference actually...
i mean i didnt feel anything extraordinary....hehe

am i killing the engine slowly by doing that?