As promised, here is my explanation of the entries in his service record.
The entry for 7 Jun 1940 seems to be the date he actually arrived back in the UK, so is consistent with him being evacuated on 6th June, technically the last day of Operation Dynamo (the evacuation of Dunkirk). As far as I am aware the RWK left France via Dunkirk, although they lost about 200 men who were either killed or taken prisoner.
Once back in Britain the battalion was assigned to home defence duties on the South Coast, and your man would have spent most of the rest of 1940 getting bored, until October when he was posted to the 8th Battalion RWK. This was a Home Defence Battalion of the Territorial Army and was largely made up of older soldiers who were perhaps past their prime when it came to fighting overseas, as well as the young recruits who could not yet be sent overseas. This was a posting because he permanently left his original unit (1 RWK) and went to join the 8th Battalion. The terminology used is that he was struck off the strength (SOS) of 1 RWK and taken on the strength (TOS) of 8 RWK.
Then within 4 months he was on attachment to the Infantry Training Centre Maidstone. 'Attached' means that he remained on the strength of 8 RWK even though he was employed somewhere else temporarily. There is something of a mystery about this attachment. The RWK's home depot was at Invicta Barracks in Maidstone and this was also the home of No 13 Infantry Training Centre, so that is quite logical. However, the mention of Brasted does not fit with this. Brasted is in North West Kent close to RAF Biggin Hill, and as far as I am aware there was never an ITC there. What I think this means is that the 8th Battalion was at Brasted (as BushInn mentions), and the ITC where he was attached was at Maidstone. Anyway, at least we know why he was there. He was learning to be a cook. The word before Cook is Learner. He then returns to 8 RWK and later qualifies as a Class 2 (Class II) cook.
Then in October 1941 he is transferred yet again, this time to 294 Company Pioneer Corps at Chattenden Barracks in North Kent. The Pioneers had been formed during WW1 but were disbanded in 1919, only to be reformed as the Auxiliary Military Pioneer Corps (AMPC) in 1939. In 1940 they were renamed the Pioneer Corps, and much later on became the Royal Pioneer Corps. They provided largely unskilled labour who assisted in particular the Royal Engineers, Royal Artillery and Royal Army Ordnance Corps. Chattenden Barracks was also home to both the Royal Engineers and a large Ordnance Depot storing munitions. Given that you man had qualified as Class II cook, that is probably how he was employed in 294 Company.
The last entry on the page, which has been cancelled, indicates that he was granted leave from 29.11.41 to 5.12 [41]. The location seems to be Milford Haven, but given that the entry was cancelled, it may have been entered in error on the wrong record sheet.
I hope this clarifies things a bit