The Soviets had a broad understanding of the Manhattan Project, though it was somewhat mixed up in places and missing big components. You have to remember that they "saw" the project through the eyes of a few spies at different places within it, so while they occasionally had areas of great detail (like the design of the implosion bomb, because Fuchs was very close to that), they also had the haziness that comes from the fact that none of those spies had the "big picture" view. It is something of the blind man and the elephant problem: lots of local detail, a sense that it ought to fit together, but missing a lot of pieces. So some of their intelligence information that tries to depict the whole project was very wrong in places: they misunderstood where the work was being done, who was running it, etc. And they were very unclear on the timeline of it.

As a result, there doesn't seem to have been a lot of influence on Soviet wartime strategy. They knew it was happening and they took efforts to get more information about it, but there wasn't much that that information could do for them in the short-term. They were also quite preoccupied with fighting the Germans for most of that time, and their resources were stretched thin.

At Potsdam, Truman alluded to the atomic bomb in a conversation with Stalin, which Stalin (correctly) interpreted as meaning that they had tested the bomb and that it was successful. He told his people that they needed to start work on their own weapon in a serious way, but it's not clear any other "policy" change came out of that. They were already planning to declare war on Japan in mid-August.

After Hiroshima, after a day of sulking in his dacha, Stalin accelerated the plans to declare war on Japan, out of fear they would surrender very quickly and he would lose the chance to gain territory.

These are basically the only strategy/policy changes that I know of that came from Stalin's information about the atomic bomb during World War II. It is worth noting that Stalin was not unique in not taking the bomb into more account — the American high command did not, either, until after the Trinity test, when the "reality" of the atomic bomb was made very evident. Truman does not seem to have factored the atomic bomb into any of the end of the war plans until after Trinity, for example.

The two main books on the Soviet atomic bomb project and its intelligence are Holloway's Stalin and the Bomb, and Gordin's Red Cloud at Dawn. The latter pays a lot of attention to these kinds of "information flows," as an aside.