In this situation, are you permitted to enter the level crossing while exercising special caution?

In the photo you are approaching a level crossing with half-barriers that are still raised; the traffic lights on both sides of the roadway are showing a flashing red signal, and another car is driving beyond the crossing. You are not allowed to enter the crossing, even while exercising particular caution - a flashing red light means no entry. The answer is: no.

The Regulation on road signs and signals states that a flashing red signal or two alternately flashing red signals mean no entry beyond the traffic lights or other device showing these signals. The Road Traffic Act requires the driver to drive at such a speed that they can stop the vehicle in a safe place when a safety device or a signal given prohibits entry onto the crossing. The red light comes on a dozen or so seconds before the half-barriers go down, when the train is already close; so raised arms do not mean that the road is clear. Particular caution does not cancel the prohibition - it is an additional duty, not a replacement. You stop in front of the traffic lights or the stop line, wait until the train has passed, the half-barriers have fully risen and the signal has gone out, and only then do you drive across.

Legal basis: § 98 para. 5 and § 99 para. 1 of the Regulation on road signs and signals; Art. 28 para. 2 of the Road Traffic Act (Prawo o ruchu drogowym)