The processor is the right socket type, however there is a chance that the motherboard BIOS will not support the instruction set of the E4400!
I noticed that the motherboard is an ECS 945GCT-HM (Livermore8), and for all intents and purposes it looks like an ECS 945GCT-M (V1.0) - the only exception being that the "-M" has both an AGP and a PCIe Slot!
Would it be possilbe to use a BIOS upgrade from the ECS website if my HP struggles when I install the E4400?
Try the e4400 in your board first without flashing the bios. You need an hp bios to run your windows installation. Even is the ecs flash works, you'll lose the ability to run your hp windows installation. Windows won't load on any non-hp motherboard unless you've already changed your windows installation to an oem or retail boxed version.
My guess is that the ecs bios won't load on your hp board. But if you've changed out a motherboard before and are confortable doing it, then it will only cost you about $50 for a low end replacement board with a g41 chipset if the flash kills your board.