Engine problems (Not Chevy)

[chevychic]

It was pretty cold last night and frosted over this morning.
When my husband went out to start up his Mitsubishi Eclipse it didn't want to start. Since it's my husband's baby, he didn't want to do anything to hurt it so that's when I jumped in and got it started since I'm not really that fond of the car.

It was acting like it wasn't getting gas and in order to get it to start I had to pump the gas a few times as I was turning the key. To keep it running, I had to continue to give it gas and keep it revved up around 1500 rpms otherwise the rpms would drop and it would try to stall. It only took about a minute of me keeping the rpms up like this until I was able to take my foot off the gas and it would run fine.

Here are some details about the car:
Fuel Injected of course
Turbo
16 valve
DOHC
2.0 Liter

One of my husband's weak points when it comes to his vehicles is he doesn't let them warm up enough before he drives them. He'll start it and let it sit for a minute or two and then he's off. I know this isn't good for the engine, have told him so, and I've been waiting for something like this to happen.
Could the injectors need cleaning? I've seen treatments that are supposed to clean the injectors but with it being a turbo I know some of that stuff can mess them up. The car is due for it's 30k mile tune up as well.

Any ideas on what could be happening and what we can do to fix it?