I've come up with this as a tentative location of the aerodrome. I've compared the few topographical features visible in the film with the few photos we have and compared them with a 1908 map of Whitfield, taking note of paths and field boundaries. I've also assumed the complex of buildings we can see on the horizon in the Pathe News Reel is the old Archer's Court Manor House, just to the north of the current Courtland Avenue. The A2 did not exist then and now cuts right through where I believe the Aerodrome, a corner of which may partly lie under TESCOs car park.
I've managed to glean a fair amount of info, but as a very brief summary, it appears that the Whitfield Aerodrome was only in existence from early 1911 to August(?) 1913. It had a brief but busy life as the staging post for competitors in Daily Mail Air races and was consequently expanded in mid 1912 including the building of a club house. The Aerodrome was run by a group known as Dover Aero Club (also formed early 1911) who were based at 11 Marine Parade. Lord Kitchener and Admiral Jellico were vice-presidents.
However, it fell victim to the August 1913 Aerial Navigation Act which closed British airspace to foreign airmen without special permissions, and extended a three mile exclusion zone around Dover Castle and Archcliffe Fort to everyone to prevent spying. As Whitfield was primarily set up to be used for cross-channel civilian flights, this effectively put an end to the activities of Dover Aero Club. Questions were raised in the Commons as to granting compensation but there is no record of whether this was granted to the owning company. Dover had effectively become a military-only fly zone. The RNAS considered adapting Whitfield, but decided against it. The RFC established Swingate instead presumably due to it's proximity to the Channel. Dover Borough Council records of March 1919 record it as disused.