hold out hope that (clause)

To "hold out hope" means to still hope for something, even though it doesn't seem very likely to happen. For example:

I sent in my application three weeks ago. They're probably not going to call me back. I'm still holding out hope, though.

You can follow "hold out hope" with a clause introduced by "that":

Kevin's family continues to hold out hope that their missing son will be returned safely to them.

This phrase appears in these lessons: