Okay, it's Power HQ, not faction home system. Faction home system means something else entirely in Elite: Dangerous.
And it depends on the Power and the activity you want to do. For outbound fortification, yes, you can only collect those supplies at HQ. For inbound fortification, you collect the supplies in any Control System and haul them to HQ.
For Preparation, you either collect in any Control System and deliver to your desired uncontrolled system or any system and deliver to any Control System.
Cargo Expansion likely works by the same rules as Preparation.
Combat vouchers (undermining, opposition, expansion) only need a Control System to be delivered at.