Is it permissible to take half an unpaid leave of absence, by agreement with the direct manager and colleagues at the hospital, where one only signs in and then leaves, due to the inability to work after the death of one's mother?

The reference for taking or not taking leave reverts to the regulations governing work in the institution. If these regulations authorize the manager to grant leave in the mentioned case, then there is no harm in it. However, if the regulations do not include this, and the matter is merely a favoritism, then it is not permissible to take leave and act in this manner, due to the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions, except for a condition that forbids what is lawful or permits what is unlawful." If there is no way but to continue working fully, then seek Allah's help and ask Him for success, and frequently remember Him; for by this, the heart is strengthened and the soul is tranquil, as Allah Almighty says: "Those who have believed and whose hearts are assured by the remembrance of Allah. Unquestionably, by the remembrance of Allah hearts are assured."