What is the ruling on paying a sum of money in exchange for staying at home with the manager's knowledge, and what is the ruling on the money earned from that? And what is the ruling on taking a "fictitious itinerary" and giving some of the day's wages in charity, given the need for the salary and the inability to bear extra work due to the care of the husband, children, and elderly parents?

We cannot definitively rule on the specific case of the questioner, but we can state the following in general terms:

Firstly: If an employee works from home with the permission of their employer and within the scope of their job duties, there is no sin upon them. Otherwise, they must adhere to the contract and fulfill its terms, based on the Almighty's saying: {O YOU WHO HAVE BELIEVED, FULFILL [YOUR] CONTRACTS}.

Secondly: It is not permissible for an official to accept a bribe to allow working from home, nor for an employee to pay it, except in cases of necessity or if it is their right and they cannot obtain it except through a bribe.

Thirdly: If the fictitious itinerary means that the employee does no work but rather pays a bribe to the official to be absent, then this is an evil act and impermissible. The salary in such a case would be consumption of public funds by wrongful means.

Fourthly: Unlawful money is not permissible to consume except for the one in dire need, based on the Almighty's saying: {AND HE HAS EXPLAINED IN DETAIL TO YOU WHAT HE HAS FORBIDDEN YOU, EXCEPT THAT TO WHICH YOU ARE COMPELLED}. Necessity is defined as a situation where one is highly likely to face destruction or unbearable hardship.