A clutch kick in modern car terminology is when you quickly disengage then re-engage the clutch with a quick kick to the clutch pedal. This allows the engine to rev momentarily, then when the clutch re-engages it will spin the wheels and induce oversteer.
A clutch kick is normally done intentionally in order to spin the car around on purpose, or otherwise get the car sideways on purpose, for example in rallying when being sideways is a more effective way to get around a corner, or more recently in drifting where being sideways around the corner is a core part of the sport.
Another way to initiate intentional oversteer is with a handbrake, so perhaps in the story mentioned it was being used as an equivalent to a hand brake turn (perhaps the kids were being pursued?).
To "clutch kick out" I would interpret as suggesting they "clutch kicked" (induced intentional oversteer) out of the situation, not that the clutch itself was kicked out, as that's not possible.
If it seems like they did in fact mean that the car was kicked out of gear (the car stalled or didn't move), then the book has it written incorrectly, and what they meant was that the gearbox was kicked out of gear. The clutch would have remained engaged in that scenario, but the transmission would be out of gear.