Yes, the coolant temperature sensor is what the ECU uses to switch to the secondary ignition map. It reads the sensor at engine start and then uses that value later to choose between the normal ignition maps and the secondary catalyst heat-up maps. This hesitation should go away completely within a couple minutes of driving once coolant temp goes above 55C. It should never happen when the car is started when warm (coolant temp > 35C). However, it is possible that a corroded temp sensor connector/wiring could give faulty readings causing this system to turn on when it shouldn't.
