Did the British Empire face a lack of administrators to run its empire, considering its size? I know natives educated to British standards were allowed into the administration in many cases, but I imagine they needed at least a proportion of native Britons to keep the countries under control.
How much bigger would it have been possible for the empire to be, ignoring political conditions, for example taking China in addition to India?
Rather speculative, as it never occurred, but due to the British tendency of utilizing pre-existing local political organizations to reduce administrative difficulty. Such as in India, using the Mughal Princes and their institutions to make their rule look more legitimate and easier. Therefore, it can reasonably believed that in a scenario where Britain where to receive the entirety of China into their empire it is very likely that they would have used the existing Imperial system to their advantage by doing something like installing a puppet emperor and therefore making administration much smoother.
Not speaking about lack of administrators, but lack of troops. The external security of the Empire, especially the Far East was enough to keep planners in Whitehall awake all night.
The British Army in 1914 numbered six infantry and one cavalry division and four division overseas. This was to police a global empire. Added to this must be the British Indian Army, which also was never very large. All this contributed to a frantic juggling of battalions from Aldershot to Cairo, and the use of 'naval brigades' of sailors onshore to support an expeditionary force (like in China). Indeed, Beatty, Jellicoe, and many other First War commanders saw much combat on land rather than sea prior to 1914.
Finally, the reserve of trained manpower in Britain was never very large. The British Army was an all-volunteer force prior to 1914 and was also one through the Twenties and some of the Thirties. This led to a very small pool of reservists, compared to France and Germany where conscription classes of hundreds of thousands could be called to the colours. Again, this would result in a frantic call for volunteers in 1916, and the tragedy of the 'Pals Battalions'.
In India, the British recruited from a small number of Indians of specific communities such as Sikhs, rajputs and other 'martial races' that were seen as both more warlike and more loyal than other communities. These 'martial races' formed a tiny portion of the total population of India, and as such there was a very small manpower reserve available to Delhi as well as Whitehall.
Security in the Far East was dependent on India. It was the Indian Army that provided much of the manpower to garrison Burma, Malaya, and the Middle East. Indian forces took part in expeditions to Sudan and Egypt in the 1800s, and the Middle East in both world wars. They fought the Japanese in Burma in WWII as well. However,by the 1900s a Russian threat to India was one that outnumbered and outgunned the Indian Army, at least in the eyes of Whitehall. Britain could not pose a credible threat to Russia from the 1920s onwards and later had much trouble with Japan in WWII. Russian annexations in Central Asia pushed its frontier to Afghanistan, and British planners constantly envisioned a Russian thrust through Afghanistan to the Indus. Although Whitehall oscillated between a 'forward policy' of strengthening buffer states to keep the Russians far off and a 'rear policy' of a defensive posture in India, what both parties could agree on was an expansion of the Indian Army and the British Army.
From the 1900s onwards, the Royal Navy in the Far East also was being eclipsed by the Japanese, and Britain could not afford to compete with them and meet the German naval threat. Add to this a resurgent Soviet Union creating buffer states in Xinjiang and Mongolia and seemingly posing a military threat to India, and a rapidly modernizing China, one saw the British economic position in Asia being reduced in comparison to the rising powers. Thus we see an inability of Britain to match its rivals in the Far East militarily due to commitments elsewhere (Germany) or economic inability (1920s & early 1930s).
As such by the 1900s British Imperial power in the East (and a lesser degree) the Middle East was being eroded relative to its rivals. This was seen by a lack of funds and political will to expand the armed forces, and a rather small reserve of manpower.
I cannot speak as to administration, though. Sorry.
Sources - 1)Srinath Raghavan India's War
2)Peter Hopkirk The Great Game
3)Paul Kennedy The Rise and Fall of the Great Powers