Is it permissible to approach a manager at work with gifts to gain their support for improving one's job situation, and is this considered lawful (halal) or unlawful (haram)?

It is not permissible to offer gifts to the manager in this manner, because it falls under bribery and gifts to workers, which are considered illicit gain (ghulul). The Prophet, peace and blessings be upon him, said: "Gifts to workers are illicit gain," and "Allah has cursed the one who gives bribes and the one who takes them." These gifts are only offered out of a desire for the manager's favor and to influence him. Islamic law has absolutely prohibited gifts to employees. What confirms this is the saying of Umar ibn Al-Khattab, may Allah be pleased with him: "Gifts to workers are illicit gain."