Let's start with Finland's war aims in 1941. Finland invaded the Soviet Union with its ally Nazi Germany in June 1941, and had reached its pre-1939 borders by August. With the war going so well, there were calls from President Risto Ryti (with egging on by German attaches such as Chief of Staff Alfred Jodl) to declare a "Greater Finland" encompassing not just its reconquered territories, but also eastern Karelia, northern Ingria, and the Kola Peninsula. This would give Finland a significantly shorter, more defensible border with the Soviet Union. Supporters of the plan came up with the slogan "Short border, long peace," though there was opposition in the Finnish parliament as well. It came to the point where a propaganda booklet, Finnlands Lebensraum, was produced by a German publisher, who added National Socialist ideology to the text. On 10 July 1941, Field Marshall Carl Gustaff Emil Mannerheim wrote the Sword Scabbard Declaration, pledging to liberate Karelia in its entirety. So there was certainly a strong political impetus for a Finnish assault to take Leningrad. Furthermore, strategically, the assault would have eliminated the entire Leningrad Front (a front being the Soviet equivalent of an army group), freeing up German and Finnish forces to continue the invasion elsewhere.

However, Finland never made it beyond approximately 20 kilometers from the Leningrad suburbs, at its old border on the Karelian isthmus. Why this happened is very complex. First of all, on 12 July 1941, the United Kingdom made an agreement with the Soviet Union and attacked Finnish ships by air at Petsamo and Kirkennes. Then, on 25 October, the United States (still neutral, but supportive of Finland's anti-communist stance) insisted that Finland end hostilities with the USSR and withdraw to its 1939 borders. The British made a similar demand in late November. In both cases, President Ryti publicly rejected the demands, but privately told the US and the UK that Finnish offensives would be halted. This is reason #1: Finland had no desire to be at war with the rest of the Allies, and so did not proceed any further with its invasion in order to prevent said war. Unfortunately, they failed with that: private guarantees did not satisfy Winston Churchill, and Britain and the rest of the Commonwealth declared war on Finland on 6 December, though this was mostly a show of support to the USSR, as Britain did not conduct any further attacks onto Finnish territory.

This brings up a broader point, though, which is that Finland was not capable of conducting an offensive war on its own. The Finnish army totaled at most 500 thousand men in 1941, possessed no armored forces other than the few (mostly outdated) Soviet tanks they captured, and had only a few hundred aircraft. There was no conceivable way they could capture Leningrad on their own, and by late 1941, its co-belligerent Germany was pushing all its forces towards Moscow. Finland was hamstrung politically and militarily. There was no way they could conduct the assault alone, and, if they did, they would invoke the (further in the case of Britain) ire of the rest of the Allies.

However, it is important to note that this situation would not arise until later in 1941, and while this would ultimately render the Finnish Army static for the rest of the war, there was still an on-going war of motion in August and September. Despite politicians carving up the maps of the Baltic and White Sea, the general attitude amongst Finnish soldiers was that this was a war to reclaim lost territory, and by August they had done that. Furthermore, in doing that it had taken heavy losses, again because of lack of air superiority and mechanization. The government also shifted to this view as it realized that a continued invasion would potentially invoke an opening of a second front by the British and the Americans. Marshall Mannerheim himself, despite his earlier declaration, had spent a great deal of time in Leningrad (when it was known as Petrograd) during his time in the Russian Imperial Army, and may have had some love for the city, which prompted his decision to halt the Finnish offensive. Furthermore, Finland, importantly, saw itself as a co-belligerent with Germany, not its ally. This meant that Finland did not want to intrinsically tie its war goals with those of the Germans. There was little desire amongst the Finnish people to make unnecessary, bloody attacks to take territory they did not want. Thus, for all of the reasons above, in 1942, Mannerheim rejected Hitler's plea for a Finnish attack to try and break the siege. An assault on Leningrad by the Finnish Army, while a bold strategic move, would have made the war significantly worse in many ways for Finland.

For sources, I will recommend:

Clements, Jonathan (2012). Mannerheim: President, Soldier, Spy. Haus Publishing.

Glantz, David (2001). The Siege of Leningrad 1941–44: 900 Days of Terror. Osceola: Zenith Press.

Vehviläinen, Olli (2002). Finland in the Second World War: Between Germany and Russia. New York: Palgrave.