Right, “공간을 주다” wouldn't work, as the “space” you mean is a bit of a metaphor Korean doesn't have. Korean might use the word “time” instead (interesting, don't you think?). There are several I can think of for those senses of the word. I'm not sure if these are what you wanted, though.
A time to think about one's relationship ― 생각할 시간
“Should we keep this relationship on or let it go?”
생각할 시간을 갖다
To have a graceful time to think (about one's relationship)
생각할 시간을 주다
To give (someone) a graceful time to think (about one's relationship)
생각할 시간을 갖게 하다
To let (someone) have a graceful time to think (about one's relationship)
A graceful time to make a decision ― 생각할 시간
“Take your time, and go with what's good for you.”
생각할 시간을 갖다
To have a graceful time to think, to make a decision.
생각할 시간을 주다
To give (someone) a graceful time to think, to make a decision.
생각할 시간을 갖게 하다
To let (someone) have a graceful time to think, to make a decision.
Me-time ― 혼자만의 시간
“From time to time, I have a me-time for my own happiness.”
혼자만의 시간을 갖다
To have a me-time, during which they can do whatever they want to, alone.
혼자만의 시간을 주다
(Nope! Rarely used.)
혼자만의 시간을 갖게 하다
To let/have (someone) have a me-time, during which they can do whatever they want to, alone.
Physical space ― 자리
자리를 내주다
To give (someone) a place by leaving the place for them (Let me know if it sounds off.)