▲句中 saying 的逻辑主语是Vladimir Putin。

Vladimir Putin said Russia would start withdrawing the “main part” of its forces in Syria, saying their main objectives (officially, fighting terrorism; actually, keeping Bashar al-Assad in power) had been achieved.

翻译：弗拉迪米尔·普京说，俄罗斯将开始撤出其在叙利亚的“主力部队”，称他们的主要目标（官方的说法是打击恐怖主义，但实际上为了保持巴沙尔-阿萨德的权力）已经实现。

▲为了避免出现 said…saying 这样的重复表达，原句也可以考虑把 saying 改为 for（表原因）。