Car hesitates when cold

Several weeks ago my wifes car(2003 Dodge Stratus SXT 2.4L DOHC) had trouble starting and then it was having trouble accelerating on her drive to work (her drive is just under 2 miles). It was fine in the afternoon. We had a problem with it hesitating a couple of years ago that turned out to be a bad spark plug. After this happened a couple of weeks ago I replaced the plugs, wires, and air filter. The car seemed to be running fine and then it started again this week. We realized it was happening only on cold mornings and once the car was warmed up it ran fine. I was doing some googling and found someone with a similar problem that said it was an air leak in a hose and the reason it ran fine after there car warmed up was because the hose would expand when it warmed up and seal the leak. I did see a split at the end of the hose coming off the air filter.
This is a picture I pulled off a website but it's the same engine my wife has. I've circled the hose that has the split in red. I wanted to find out if people thought that was a reasonable explanation for why the car hesitates when it's cold. I also wanted to find out what I call that hose so I can try and find the part to order. The hose goes up into the engine from the air filter.

Thanks for any advice anyone can offer.

That looks like it should be the PCV valve hose...if the 03 2.4L has that sort of set up (I'm an old school engine type). A split in that hose could cause hesitation.