What does Emilia's remark about the rift between Othello and Cassio suggest about their relationship?
I'm not exactly sure what quote you mean. Is it this one perhaps?
Good morrow, good Lieutenant: I am sorry
For your displeasure; but all will sure be well.
The general and his wife are talking of it;
And she speaks for you stoutly: the Moor replies,
That he you hurt is of great fame in Cyprus,
And great affinity, and that in wholesome wisdom
He might not but refuse you; but he protests he loves you
And needs no other suitor but his likings
To take the safest occasion by the front
To bring you in again.
In Othello Act III, Emilia summarizes the closed-door "trial" of Cassio here. Just as Othello battled Brabantio in court in Act I and won, so too does he battle Desdemona and win regarding the plight of Cassio.
Emilia hopes that her husband too will be a kind of "lawyer" in the matter:
Desdemona tries to argue that Othello loves Cassio, but Othello responds that he must protect the status and reputation of the wounded man, Montano. In his decision-making, Othello values reputation above "love." In short, Cassio wounded Montano AND Othello more in terms of reputation than in bodily harm. He must, therefore, be punished by a demotion in rank and reputation.
Later, after Emilia steals the handkerchief, she will tell Desdemona:
After Iago has used her for getting the handkerchief, Emilia will lose her optimism that either Desdemona or Iago can talk Othello into forgiveness. She believes that men are consumers only, of women and men of lower status.
It is clear that Emilia's remarks show that Othello values his public reputation above personal relationships.