Is the financial sum provided by one of the companies to help the questioner's needy friend considered a bribe, knowing that the questioner's dealings with companies are conducted with complete integrity?

It appears that the amount is a charity for the sake of Allah, given by the person to the sick woman. It is not considered a bribe because the questioner is not benefiting from the money. We hope that the questioner will be rewarded for guiding to good, and the person for helping those in need.