Take some time off work

I would probably write something like this:

Date

Dear Manager (Use manager's name e.g. Mr Smith)

I wish to apply for leave of absence to care for my mother, who is sick in hospital. As the only child of the family, I am the only person available to do this. I can provide the telegram I have received from the hospital in support of this application if required. I am prepared to make up for any missed work upon my return.

Your respectfully

Mary

Thank you. The changes you made seem very natrual. And is the reason why you deleted the last sentence wordy? And please help me compare A with B. Which one is better? And could you make more changes to the the better one? Thank you.
A
Dear manager,
Please allow me to take 2 days off work for something urgent. The department I rent expired yesterday. And my landlord asked me to move out in 3 days for there is another renter will move in. so I have to find a new department and make a move as soon as possible. If there are extra arrangements for me during the 2 days, you can tell my partner Lily about them. I will finish them as quickly as I can after my day off.
B
Dear Mr. Smith,
Please allow me to take 2 days off work for an emergency. The apartment I have rent will expired in a couple of days, however, yesterday I was required to move out in 2 days for the landlord had rent it to another one. Therefore, I have to rent a new one and move to it as soon as possible. If there are extra arrangements needing me to deal with during my absence, please transfer to my partner, Lily, whom I will hand over to, or suspend them until I come back to work.