Hi there. I just completed a kill mission against a bunch of Authority, but there's no mention of where the mission needs to be cashed in or even who I'm working for.
From the left hand HUD panel the second tab labelled "transactions" (from memory, not in game right now), and highlight the specific mission should give you a pop-up with more detail of what you took and where. I recall from Beta testing there was an issue on "take this cargo" missions that you needed to return to the starting system to collect the cash, but this wasn't listed on the mission. This is now changed and it's cash on delivery.
I've not taken a "kill" mission, so am unclear on the mechanics, sorry.
Incidentally, how do you know when the game considers you to have killed enough of the target ships? Do we just assume or guess? Because the factions aren't terribly well-defined.