I took my car to get a tune up recently and the mechanic said it needs the O2 sensor and maf replaced. The MAF sensor alone could make quite a difference and it is easy to change out. Try disconnecting the MAF wiring harness and see if the enigine starts better. 2004 Volvo V40 Mass Air Flow Sensor ...

I think you got more going on then what the mechanic said. I suspect it is not getting enough air ... when you push down on the throttle your giving the engine more air, the engine control module decides how much fuel. Try removing the air filter see if it makes a difference but do not leave it that...