I suppose the focus of the later part of 1942 becomes the defence of Hawaii and proetcting the SLOCs to the West Coast.
Focus would be on defending Hawaii, but that is the responsibility of the USAAF and USA; USN carrier assets would probably focus on deploying to the central Pacific on airstrikes like they did both before and after Midway but able to retain the ability to ambush any IJN forces deployed to attack the Hawaiian Islands.
SLOCS to the West Coast are pretty secure, the focus would be on reinforcing the SLOCS to New Zealand and Australia; expect more reinforcements for Samoa, Fiji, New Caledonia and French Polynesia.
How would the eventual USN offensive occur, in this TL? Would RN try and take the offensive in the Far East to relieve the pressue on the Americans.
Definitely no USMC landing on Guadalcanal in August 1942 and no Solomons Campaign in 1943. USN would focus on raids around the periphery of the Japanese defensive perimeter.
First batch of Essex carriers probably can't be deployed much sooner than OTL; the second batch that arrived in OTL late 1943/early 1944 could be probably be deployed 2-3 months earlier.
IJN ability to expand much further than OTL conquests is pretty limited; they had Operation FS on the books but it's doubtful they would have attempted it even if US carriers force was destroyed. Just not enough assets to see it through. I could see the Japanese make renewed efforts to capture Port Moresby and evict the Australians from New Guinea. There is really not much the USN could do to help the Australians with a severely reduced carrier force.
I would think that the US would begin significant offensive operations in the central Pacific in the late 1943 timeframe, fairly similar to what OTL occurred. The question mark would be would the US consider going straight at the Marshall Islands and would it first would to test its amphibious doctrine elsewhere such as the Gilbert Islands, ie, Tarawa. With the reduced carrier assets, I don't see the two-pronged advance from both the central and southwest Pacific theaters, King would ensure that the Central Pacific advance would be the main effort.
Would RN try and take the offensive in the Far East to relieve the pressue on the Americans.
I just don't think the RN has the assets to do much in the Far East and I doubt that the Japanese would take these efforts seriously. The British have their hands full with the Med and keeping the sea lanes to Great Britain open. They just don't have any surplus capacity to do much in the Far East in the second half of 1942.