The large hose leaving the turbo and going forward are inter cooler (IC) hoses. The intake air goes through the turbo, then through the large hose to the intercooler in front of the radiator, out the bottom of the IC and back and up to the throttle body.

Any leak in the IC system will result in no turbo boost. A large leak, like a loose hose will result in no power at all and cause most people to park on the side of the road wondering why it idles just fine.

I wouldn't worry about the oil mist in the IC. As the oil seals in the very fast spinning turbo wear, oil will make its way to the air supply side. This is quite common and of little concern, especially in high mileage engines. But it does mean that you will be consuming about 1 quart of oil every 2-3K miles, depending on turbo usage. You will also see a small drip of oil under the front of the parked car, about 25cents size. That oil comes from the factory installed weep hole in the bottom of the IC, put there so that excess oil from the turbo has some place to go rather than filling up the IC and blocking air flow.

If you haven't yet, find a pick & pull junk yard and pull off the IC hoses from any turbo Volvo, 1994-2000 as they are all the same. These hoses are normally very robust and not prone to failure. Clean the inside of the 'used' hoses with soapy water and let dry.

As fourvolvos says, a hole in the upper intercooler hose is not good and will not permit pressure to build. Also, you say you can see oil vapor emitting from the hole which sounds to me like you have oil seals going bad on the turbo. Which softened the rubber hose eventually leading to failure.

Me? The hose is cheap enough, replace it. Then decide on whether to look into the source of the oil vapor. But the problem is not going to get better on it's own.
--
"Differences of opinions should be tolerated, but not if they're too different' - Sharon Craig

The rubber hose (elbow) that you are looking at is the upper turbo hose and if it has a hole in it there is your problem. It connects the inter-cooler to a pipe that feeds into the throttle body.
Not an expensive part.
http://www.eeuroparts.com/Parts/51171/Intercooler-Hose-upper-9161092M/?diagramId=1164&diagramNumber=3
You could wrap it with good duct tape (temporarily) to check it out while you order the replacement.
Good Luck,
Tom