" I have to go home so I need to leave early" is it correct when we are asking for early leave permission in mail.
If this is all you say, it might be considered rude. It can sound like you expect or are entitled to leave early. Also, it seems vague. I would recommend that you include the reason that you need to leave, if possible. I understand that the reason might be a private/personal reason, but the person from whom you request permission might require some proof.
You might consider
- I have to go home so I need to leave early. Is this (that) ok?
- I need to go home because (reason). Is this fine?
- (Reason), so I need to go home early? Is that ok?
- (Reason). May I leave early (on date you need to leave early)?
I have to go home so I need to leave early
Grammatically it's fine although I'd add a comma after home.
However, on it's own it may not be enough. In most work environments you would be expected (and it's more polite) to provide more information - Why do you have to go home? What do you mean by early?
Consider something like the following:
A plumber is coming to fix my shower at home, so I need to leave by 2pm
It depends a lot on the culture where you are making the request