In this situation, are you allowed to use the horn in order to induce the driver of the preceding vehicle to leave the intersection faster?

In the video you are driving along a street in town behind a red car, which has stopped with its brake lights on just after a pedestrian crossing, before a junction, letting through vehicles coming from the opposite direction, including a yellow car; by the road there are signs B-33 "40" and D-6 "pedestrian crossing". You are not allowed to use the horn to hurry the driver in front of you - the horn is not for rushing others. The answer is: no.

The Road Traffic Act allows a driver to use the horn only when it is necessary to warn of danger, and prohibits overusing the horn and using it in a built-up area unless it is necessary because of immediate danger. The driver of the red car is waiting, most probably for a chance to turn left, and is giving way to vehicles from the opposite direction - this is not a danger, but correct behaviour, and sounding the horn could stress them and make them move off suddenly and dangerously in front of an oncoming vehicle. You stop behind them at a safe distance and move off when the junction is clear.

Legal basis: Art. 29 para. 1 and para. 2 point 1 and 2 of the Road Traffic Act (Prawo o ruchu drogowym)