What's the relationship between Carter and Hanaway?

If you ask about the relationship between them within the movie, there are a few no of proofs which led us to believe that they might have been in a relationship. The points are:

1) Like you posted, Carter seemed particularly broken up with the death of Hannaway. She was very sad to see him die.

2) Before the death, Hannaway said:

You are so beautiful. I am sorry I should have told you.

3) After the killing of Sabine Moreau, Brant argued that throwing her off the building was an instinctive move by Carter. He suspected she took the revenge of the death of Hannaway by killing her.

If you ask about if they were having any relationship between them off screen, I would say neither did I hear about it, nor Google could cast any light on this! :P

I have not yet seen MI:RN, but the first two films in which JJ Abrams and [his] Bad Robot Productions were hired to help revitalize the franchise. Both films MI:III & Ghost Protocol feature a tragic cameo from well-known Bad Robot TV show alumni, whose relationships with another main character come off as "unrequited" and whose true nature is often unclear.

Kerri Russell is the lead actress from pre-Bad Robot series Felicity and she plays agent Lindsay Farris whom tragically meets her end in Mission Impossible: III. Ethan Hunt was her trainer. He tells Luther that she was like a little sister to him, but in truth we do not know if it was possible, if one of them did have romantic feelings for the other--and/or if it just was never returned, but it is clear Ethan moved on from Lindsay and has remained loyal to Julia.

Josh Holloway is one of many lead actors who was main character James "Sawyer" Ford on all six seasons of LOST. Agent Trevor Hannaway and Carter's relationship then perhaps mirrors Ethan's and Lindsay's, as Jane Carter is now in the role Ethan "was" in the previous installment and that makes these cameos also thematic as a kind of echo running through the films...

In both of these cases, these main characters were also "hearthrobs" and often apart of love triangles or love squares...

Like I said, I haven't seen Rogue Nation which seems a bit more of McQuarrie's baby, to know if there are any Bad Robot references or cameos.

However, I do know that MI:VI will feature actress Angela Bassett in a role that seems rather reminiscent to her [Bad Robot] role CIA Director Hayden CHASE on Alias, as MI:III and Protocol has a TON of Alias references and allusions in it!