" He finally resigned from the Cabinet when senior army commanders were given free access to the Cabinet"

If this is indeed true then there must be a date for this resignation.

His only appointment was that of Secretary of State for War so that must have been the Cabinet Post he resigned from. Any idea who his successor was?

The official dates for those holding the post of Secretary of State for War during the Great War are as follows:

Herbert Asquith - Secretary of State for War 30.03.1914 to 5.08.1914

Herbert Kitchener - Secretary of Sate for War 5.08.1914 to 5.06.1916 (Died as a result of enemy action)

David Lloyd George - Secretary of State for War 6.07.1916 to 5.12.1916

Earl of Derby - Secretary of State for War 10.12.1916 to 18.04.1918

Viscount Milner - Secretary of Sate for War 18.04.1918 to 10.01.1919

I would suggest that The History Learning Site article linked to is wrong and the information given is incorrect. Anyone wishing to refute that statement please come up with the date of Lord Kitchener's resignation.

Look at the gaps between each succeeding Secretary of State for War - either immediate hand-over or at most a few days delay, with the exception of Kitchener where the position lay vacant from the 5th June until the 6th July while a successor was sought. Which sort of pours doubt on everyone being eager to get rid of Kitchener - logically if you are keen to get rid of someone in an organisation you line up the likely replacement before you get rid of the incumbent, and that demonstrably was not the case when Kitchener died suddenly with the loss of HMS Hampshire.