Technically, no.
However, I would strongly recommend that you do, not just out of courtesy but also for traffic information. Why not use the service that's available to you? i.e. ATC, AFISO and A/G will be able to advise you of anything relating to your safety.
Don't forget (relating to your Goodwood/Farnborough query) that although F'boro provide an excellent service, Radar coverage/performance is by no means perfect and they may not be able to see all aircraft on their screen. Some aircraft are too small/light to paint on radar!
To summarise, in this particular scenario I would say it is worth checking in with Goodwood. They can confirm/update anything you know from Farnborough, and they can then tell other a/c about you!
Foxy