So it would be best to get clearance through FF rather than directly from approach?
The misuse of "clearance" aside, from this sentence I can't tell whether you understand that FF is not an ATC facility. its a service provided
by an ATC facility. The ATC facility providing FF services could be Center, Tracon (approach/departure), or even a Tower.
pi**ing contests about what's necessary or not aside, what a lot of folks in this thread have essentially said is that in the area that includes Class C airspace, the facility that provides FF services will generally be approach.
That's correct. In the normal course of events, the ATC facility that has been providing FF services will transfer you along the line as and if necessary. If you start your FF services with a Class D Tower, it should hand you off at the appropriate time to Tracon which should hand you off at the appropriate time to Center, etc.
That makes some sense because which facility is the "proper" one can depend not only on the airspace as shown on your sectional but also on how that airspace is divided up within an ATC facility or between ATC facilities in ways that the pilot can't be expected to know about. Just for example, I've landed at a towered airport while speaking with Approach Control (the "proper" facility at that time and place).
The pi**ing contest comes up when you as the pilot think to yourself, "Hmm, I've been receiving FF from Center, I'm approaching Class C and I haven't been handed off to Approach." Although ATC is supposed to coordinate this stuff for people they are talking to, sometimes they don't.
In case of doubt, I'd go with Jesse's
I would go with "center, should 81869 remain with you into the charlie?"
as the simplest and most sensible approach. As Jesse said, it's not rocket science, just an understanding of the system and some simple courtesy