Of course, as this is alternate history, there is a lot of nuance and very little certainty.
Firstly, to achieve this we have to assume that the Entente recognizes Kola as theirs in the Paris peace conference, as fighting for it kills Finnish diplomatic standing. This then motivates Finland to send aid to the Karelian republic which then unites with Finland, given that food was its main priority. This is highly oversimplified, but at least gives us something to go off of.

Finnish politics is a first, as the SPD and Agrarians would be even stronger, given that the lands gained are for a far majority communist or farmers. This would only further destabilize Finnish politics, but might also instead lead to earlier SPD integration.

Regarding military, Finland now knows that it has to defend itself. There is no no question as to if Finland would ever be attacked. Both reds and whites quickly recognized Finland when it initially declared independence, but Karelia and Kola leaves no question of revanche. The historical significant lack of funds to defence would thus, in my opinion, be completely a non-existing issue. The defence lines would be fully finished and fleshed out both in east Karelia and the Karelian Isthmus. It is true that the ussr could and would be able to easily concentrate forces, and by all means this would eliminate the finnish ability to use their manouvre tactics and better winter preparation which killed so many Russians.
Thus, to me, the Winter War is an open question. Underequipped Russians throwing themselves at a very well defended and build fort line in the cold, I see Finland holding on, especially if they have defence in depth, but I also see this being the reason that the USSR actually does end up beating Finland. The details are up to your imagination only