Possibly end the war if Finland's objective from the outset was the destruction of the Northern Front of Soviet forces. Which isn't an unrealistic objective and which leads subjectively to collapse of the Soviet northwest front and so on as elements filter southward from the 'safely' captured territory.
This is actually very unrealistic. Russia losing Kola and all of East Karelia would have been insignificant to the outcome of the war once the Murmansk railway would already have been cut anyway, with an assault in the south to Sorokka (officially Belomorsk since 1938). All that really mattered in the region that the Finns or Germans didn't already hold, was such a portion of the Murmansk railway that would have allowed cutting it off completely. On top of that, the railway itself greatly decreased in significance after 1941, and alternative routes of lend-lease would've opened sooner, had the Russians lost it.
Not only all that, but taking those territories was realistically not even possible. The infrastructure and terrain for such an operation was not there, which General Airo warned the Germans about when their military was inspecting terrain in Finnish Lapland prior to Barbarossa. Not only that, but neither Finland nor Germany had the forces to spare on what essentially was throwing men away on a fool's errand, though arguably the Germans tried at first, before giving up, opting to learn the hard way what Airo had already told them.
The campaign season of 1942 was the Finnish armies time to shine but, an undocumented collapse in strategic planning happened.
Germany had lost the war by December 1941. She was not coming back from failing to route Russia's army before the onset of winter. There was nothing more for Finland to do, except perhaps take Sorokka, cutting the Murmansk railway, which she easily could have, and then relinquished control of the area to German divisions redeploying there from the north, as to not overstretch the scarce Finnish forces too much.
Getting involved in Leningrad was both militarily and politically a high-risk, high-cost, low-reward option. Finland had leaders smart enough not to take it. Yes, Finland getting involved would maybe, IMO probably, have meant the city would have eventually fallen, but so what? It would have come at a great cost in lives and materiel, would've taken a sizable, likely primarily German garrison to police (Finns didn't have a lot of manpower to spare), it would not have saved Germany by late-'41 anyway, and then there's the question of what kind of backlash would have come from the Russians once they came back knocking: the Karelian Offensive might've been even greater, or another offensive of similar scale could have come afterwards, which even if repulsed, would've resulted in a great many more Finns losing their lives.
I used to think that maybe if Finland opted to push on Leningrad in 1941 and took it together with the Germans, that could
maybe,
possibly, have changed the outcome of the Eastern Front and by extension WW2 in Europe, by allowing Heeresgruppe Nord to continue east and then south to help Heeresgruppe Mitte with taking Moscow, as per German plans, allowing the city to fall, which in turn would have screwed up Russian supply, communications, etc.
But honestly the more I read into it I realise:
a) There probably was not enough time to take Leningrad in a timely manner to help HG Mitte take Moscow before the onset of the mud season or even winter.
b) HG Nord would probably have become bogged down east of Leningrad anyway, before getting a chance to help HG Mitte.
c) The Germans weren't really capable of taking Moscow, its surrounding region and then holding on to it in the first place.
d) Even if the Germans by some act of providence did, it's possible the Allies would simply have stepped up their lend-lease game even more and kept Russia from collapsing (the latter which they pretty much did already IRL).
So no, I don't think there's much the Finnish army could have done differently in 1942, besides not taking Sorokka.
I seriously doubt it was consideration of the US-Finnish relations. That sounds like some half baked 1980s NATO propaganda.
It was a political decision, and this is well-documented. Militarily there was no major obstacle to cutting the Murmansk railway by taking Sorokka, and it could have been accomplished with ease. Airo made a plan for it, it was displayed to Mannerheim and Ryti, and they mutually agreed against it, mostly because USA threatened to declare war on Finland if she cut the railway. The idea was that USA could keep the Russians off Finland's back post-war in case Germany lost. Of course in reality it ended up being the Finnish military that kept the Russians from Sovietizing the country, not American diplomatic pressure, so politically this strategy of not angering the US didn't amount to much, if anything at all. But 20/20 hindsight.
Probably the early 1942 Soviet counter attacks caused routing which Finnish high command feared might lead to early collapse of the army.
This accusation is baseless. The Russian 1942 counter-attacks were repulsed with ease. AFAIK some of the partly demobilised and thus understrength units south of the river Syväri (Svir in Russian) initially had some issues, but it was nothing actually threatening to the Finnish situation.
There was not a lightest sign of the Finnish Army collapsing in 1942 - but the routing and the collapse was very near two years later in 1944.
I think this is one of those things that's got some tiny grain of truth to it, but is mostly a myth that has been overplayed in the Finnish "popular history" culture. While I'm sure there were some individual cases of panicking at the start of the Russian Karelian Offensive in June 1944, by and large the Finnish military conducted an orderly retreat as the first Finnish lines of defence were breached, all the while engaged in delaying action against the Russians, until eventually halting the Russian advance on all fronts.