The Germans were absolutely to blame for the siege of Leningrad and while the Russians did everything they could to provide any support or relief, the other fronts called them away from a large scale offensive.

By the time the Germans got to Leningrad, the decision was made to starve and exterminate the population. This decision was made as not to waste time with a long battle and not to take care of a large population. When the the defenses were weakened due to hunger and disease, the Germans would strike. They were in no rush to take the city as they had it completely surrounded. The Russians did supply the city over Lake Ladoga when it froze, but during the summer and early fall, the city had ration what food it had left. The ration supply by October 1941 was 400g of bread for workers, 200g for their dependents. Meat, milk, and vegetables were in very short supply and sometimes non existent. And the wait in the cold for hours for these meager rations was excruciating. The rations were a far cry from a couple months ago which were 800g of bread for each worker and 600g for each dependent. The bread before October had less disgusting additives like sawdust, oates, burnt flour from bombed out storages, cellulose from wood. By February 1942, the effects of the siege was less devastating and people got their food on time and in larger quantities. This did not mean the siege or the casualties decreased. During the height of the siege, thousands were dying a week.

The shortest route between the city was a fortified front of 10Km between the Leningrad Front and the Northern Front. The Germans controlled this area and fortified it well. The only way supplies could come in was over Lake Ladoga. The path over Lake Ladoga was known as the Road of Life when it froze during the winter. It brought in the bare minimum of food and supplies for the city. It was a dangerous path as it was ice and not knowing where the thin ice was could lead to certain death for many truck drivers. Artillery and air attacks would constantly break the ice and sink trucks and men. Combined with infantry and tanks targeting the trucks as well. AA guns and trenches were built in the thick ice to help defend the Road of Life. The corridor you speak of was not established until 1943, two years into the siege and many casualties on the Russian side to secure it. This helped get a constant supply of food, fuel, and other supplies into the city, but it was targeted by artillery and air raids. But it was much safer than the Road of Life and could bring in more supplies daily. With the establishment of the land corridor might have helped the supply situation, it by no means ended the siege as Army Group North was well fortified and would not give up any more ground.

The Germans were willing to let the entire population of Leningrad freeze and die of hunger so it did not have to take care of the city if it occupied it. While the Russians could not spare entire armies to help lift the siege due to Moscow, that does not mean they forgot about them. They constantly supplied the city with whatever it could and when it could launch an offensive, it was when it was strong enough to do so.

Addon: In the winter of 1941-42 the rationing got so bad that workers only got 125g of bread and their dependents got none. Reports of cannibalism were said to have been the worst around this time. People would even boil wallpaper as it might have some cellulose left.

Source:

Glantz, David M. The Siege of Leningrad, 1941-1944: 900 Days of Terror. Cassell Military Paperbacks, 2004.