Is it permissible to accept promotional gifts (immunity-boosting medicines and bee honey) offered by a private company through one's superior, knowing that the beneficiaries cannot show favoritism towards this company? And if it is not permissible, what is the required action regarding what has been used and what has not been used?

If the governmental company's regulations permit accepting gifts from companies it deals with, then there is no harm in benefiting from them. However, if the regulations do not permit it, then it is not permissible to benefit from them, and they must be returned or their value donated if returning them is impossible. If the regulations do not permit accepting gifts, they should be returned to the gifting company. If they do permit accepting them, but your superior is not authorized to distribute them, they should be returned to the governmental company.