I have a temporary residence card with the annotation "STUDENT". Do I have access to the labor market?

YES. The annotation "student" is placed on the residence card in the case of a permit granted for studies, i.e. pursuant to Art. 144 of the Act on Foreigners.

However, a foreigner who holds a temporary residence permit in the Republic of Poland granted in connection with the circumstances referred to in Article 144 of the Act on Foreigners is exempt from the obligation to have a work permit.