The image of falling snow is meant to illustrate the "...winter planet..." mentioned in the opening sentence of "The Game of Glory." See Poul Anderson, Captain Flandry: Defender Of The Terran Empire (New York, 2010), p. 303.
Flandry is indeed lucky in what becomes l'affaire Nyanza. The Merseian agent, A'u, has:
"...fled Conjumar in a poisoned wreck of a spaceship, which might have gone twenty light years before killing its pilot..." (ibid.)
In a sphere twenty light years across, Flandry's men predictably find no trace of a being who must be lying low. However, two Earth-years later, while Flandry is directing Intelligence operations during the suppression of Brae, also in the Spican province, he and his escort pass near a bushwacked squad of marines. The single marine hit by the sniper, Thomas Umbolu, dies in Flandry's arms, saying just enough to suggest that someone might be inciting disaffection on his home planet.
Tom says:
"'It's him in Uhunhu that knows...shall freedom come from slave-masters, asked he in Uhunhu. He and his 'ull teach what we must know...'" (p. 305)
Investigation reveals that Tom is from a colony planet named Nyanza a mere five parsecs away where there is indeed disaffection - the Imperial resident has just been assassinated - and that Uhunhu is "...a permanently submerged area..." (p. 322) in the turbulent Nyanzan ocean. A'u is amphibious. Two investigations converge.
Everything that Flandry does after hearing Tom's dying words is down to his own skills and meticulousness as an Intelligence agent. He is well placed to take advantage of the luck, from his point of view, that Tom was shot where and when he was and that he spoke before he died.
