Not sure if this is the right forum, feel free to move.
I could have sworn I've read in discussions here that despite their stated aims of adjusting the border, the Soviet Unions actual aim was the (re)assimilation of Finland, and that they actually had a puppet government ready to install. But I've been unable to find references to this elsewhere.
Does anyone remember anything about the stated aims of the Soviets being different from the actual ones?
As far as I know, there is no direct evidence available to say that the USSR wanted to annex Finland, as in official documents with Stalin discussing this with other members of the Soviet leadership. No such smoking gun has surfaced so far.
There is plenty of circumstantial evidence to point to the USSR wanting to conquer all of Finland and at the very least make it into a puppet state, though. The puppet government you mentioned is of course the so-called Kuusinen government or Terijoki government. Wikipedia calls it
the Finnish Democratic Republic. From the beginning of the actual hostilities in 1939, the Soviets' stated aim moved from border adjustments to helping the "real Finnish people's government" to oust the "White bandits" of the "junta" leading Finland. For most of the war, the USSR actually did not recognize the legal Finnish government of President Kallio and Prime Minister Ryti but considered its own puppet the legitimate Finnish government it had diplomatic relations with. This was openly and officially told to all foreign powers, the League of Nations, etc. But in January 1940 Moscow de facto ditched this conceit and "re-recognized" the bourgeois Finnish government by starting to negotiate with it again.
Other information pointing towards the aim to conquer all of Finland was, for example, the guide book on Finland issued to the Red Army in the run-up to the war that included specific information about all the parts of the country with helpful specifics like how not to mistakenly cross over to Sweden at the Finno-Swedish border and while there, to remember to be courteous to the Swedish border guards... Also, when we look at the Molotov-Ribbentrop Pact's secret protocol (in its amended form), with the exception of Finland all the other areas placed in the Soviet sphere of influence in 1939 were annexed directly into the USSR by 1941.
Given especially the fate of the Baltic states, which also got puppet governments that then "voluntarily" asked to join the USSR, it seems quite likely that a similar fate had been prepared for Finland had the Red Army completed the conquest of Finland by the end of 1940 like was planned. Without the Finnish government's refusal to hand over border areas to the USSR in the fall of 1939, the likeliest outcome for 39-40 would have been a similar creeping takeover for Finland like happened in the Baltic states that did accept the Soviet demands made in the same timeframe.
