"fighting hard" is correct and usable in written English. You can use it to describe someone who is putting a lot of effort into something, or to describe a situation where two sides are involved in a difficult conflict. Example: "The students were fighting hard to make sure their voices were heard during the meeting.".
We're all fighting hard and I've got to continue to fight hard".
They're going to be fighting hard".
So Mr Llewelyn is fighting hard.
But Mr Bush's people are fighting hard.
A band of brothers fighting hard wars.
And Mr Rumsfeld is fighting hard for his political life.
"I'm just worn out after fighting hard all day.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.