2005 Galant start the car and runs rough on idle and when I press on gas bogs down real bad. last time it was in shop the replaced a cat said it was clogged you think it could still be a cat problem?

SOURCE: 2000 mistubishi galant es idling rough seriously

Idling problems may be cause buy a number of issues.  You may check out the fast idle control valve and PCV if they would need cleaning.  You can check out the timing it may be advanced, that is why you feel that the car runs hard.

SOURCE: 1996 Galant runs rough until gas pedal pressed

You should have your TPS and EGR looked at as those cause rough idles...Other causes are bad idle air control valve, bad plugs, bad distributor cap & rotor, a bad wire, bad O2 sensor, etc...

Testimonial: "Thank you for your time. We have replaced the plugs,wires,rotor,cap and distributor.If TPS,EGR,o2 bad,wouldnt engine light come on?"

SOURCE: 2001 Galant service engine light on & won't stay running/ runs rough

Try this: Disconnect the ecm control (where the air filter is housed)
Let me know if that worked.

SOURCE: I have a 1997 Mitsubishi Galant with rough idle

I am having the same issue on Mitsubishi Galant 02. Clattering on acceleration, herky jerky idle. Pulled a code multiple cylinder misfire. My mechanic's diagnosis is bad valve cover gasket causing oil to seap into the spark plugs and bad timing belt, which I have not replaced at 150k miles. I do however know my fuel injectors do need cleaning, he hasn't ruled that out either. Have you pulled any code on it? I have read on engine knock sensor going out on some of these also. Hope this helps some.