To deal with any inconsistencies in the speed at which they complete the 58.5 seconds. If a particular defective clock takes 59.9 seconds instead of 58.5 it'll be just fine, as it will start going again when it gets the next minute pulse from the central system.
Because then any slow clocks have 1.5 seconds to receive additional pulses from the master clock to bring them into sync.
There are no additional pulses. Only minute pulses. It's a motor driving the second hand. The motor starts by the minute pulse and stops itself when reaching the top.