MOSCOW—A second consecutive weekend of mass detentions following protests for free elections in the Russian capital left the opposition undeterred and President Vladimir Putin’s ability to manage growing public discontent in question.

Around 1,000 people who had braved the rain were detained on Saturday, according to independent observers, a week after more than 1,300 people were held and dozens injured in similar demonstrations. The opposition said it is planning new protests for this coming weekend.

The...