The correct spelling is furlough
The pronoun case of the bolded word <em>him </em>is objective case.
The words <em>to him </em>are used as indirect object in this sentence, which means that the case that the pronoun is used in is the objective case, or the accusative, in technical terms. The other case would be the subjective case, but given that there is no obvious subject in this sentence (it is implied), there are no subjective cases.
If in email or message online
hello _____,
I’m sorry i have not shown up to class as often as i would have liked in the last week. I have came down with a cold and it’s been really difficult for me to finish some of the work i did most of it but am having trouble completing all of it. I will try to make it back to school soon but i doubt i’ll be able to come tomorrow. When i get back to school can we talk about a possible time and date i could do the test so i don’t fall fall behind. Sorry for the inconvenience.
sincerely______