Rav Huna ruled that it would be permitted to bake for the soldiers if the bakers were permitted to give bread to the Jewish children who were around, as well. In such a case, every loaf of bread could be seen as potentially being baked for the children. If the soldiers were careful that none of the bread be given away, and insisted that it all be delivered to the soldiers, then it would be forbidden to bake for them.
The Gemara challenges Rav Huna’s lenient ruling: But isn’t it taught in a baraita: There was an incident involving Shimon the Timnite, who did not come on the night of the Festival to the study hall. In the morning, Rabbi Yehuda ben Bava found him and said to him: Why did you not come last night to the study hall? He said to him: A military unit on a search mission [balleshet] came to our city and wanted to pillage the entire city. We slaughtered a calf in order to placate them, and we fed them with it and had them depart in peace.
Rabbi Yehuda ben Bava objected to this story, pointing out that the passage permitting cooking on Yom Tov (Sh’mot 12:16) only allows it lahem – for you – not for non-Jews. As the Gemara explains, in this case the animal that was prepared for the balleshet was not kosher, so it could not have been eaten by Jews and the entire preparation was for non-Jews only.
The term balleshet apparently refers to an army unit that was sent to search for valuables (the root b-l-sh means to search). Usually these units were employed in enforcing payment of taxes, which made it essential for the local communities to stay on good terms with them, since their broad mandate often allowed them to stray well beyond their official tasks into violence and looting.