Complex sentence: Over the years this office was to quietly but steadily claim the power of government over at least the Russians in the region--taxation, policing, administration, and justice--powers that should have belonged to, but were effectively relinquished by, the Qing.
Split into simpler sentences: Over the years this office quietly began to unofficially govern at least the Russians in the region, managing taxation, policing, administration, and justice. These powers officially belonged to, but were effectively relinquished by, the Qing.