The interview between the Duke and the gentleman representing the Count to discuss money matters is terminated quite abruptly near the end of "My Last Duchess" when the representative apparently jumps up and starts to leave the room to descend the stairs without saying a word. These are the pertinent lines from the poem:
Will't please you rise? We'll meet
The company below, then. I repeat,
The Count your master's known munificence
Is ample warrant that no just pretence
Of mine for dowry will be disallowed;
Though his fair daughter's self, as I avowed
At starting, is my object.
The Duke has been explaining what he disliked about his last duchess, a beautiful girl who was loved by everyone and whom he has apparently had murdered. He has unwittingly revealed his own character as that of a cold, selfish, greedy, insensitive monster who values the painting of his dead wife more than he valued the girl herself. (Note how Browning creates an impression of the Duke's insensitivity with the crudeness of the open couplets that comprise the entire poem, e.g., "meet" and "repeat," "munificence" and "pretence.")
The Count's representative surprises the Duke by jumping out of his chair and retreating from the room with the intention of starting down the stairs to where, presumably, the Count himself and some members of his entourage are assembled. The Duke hurries to catch up with the retreating visitor because he does not want to have the impression created that their meeting broke up with disagreement and ill feelings. He invited the other man upstairs on the pretext of showing him his art collection, but the question of the dowry was uppermost at the beginning and ending of their interview. The Duke can go downstairs and prevent the agent from talking to his master in confidence, while in the meantime he can preserve the fiction that they only went upstairs to look at some paintings and statues.
The agent's sudden and rude departure from the upstairs room strongly suggests that he intends to advise the Count against marrying his daughter to this heartless, tyrannical suitor under any circumstances. Since the question of the amount of the dowry was never settled, the Count cannot go ahead with discussing other details of the proposed marriage. He and his party, probably including his wife and perhaps even his daughter, will leave for home with cold formality and words of farewell alluding to the occasion as nothing more than a friendly visit.
The Duke must realize he has made a terrible impression, and if you were to write a letter on his behalf you would have to try to salvage the engagement to the Count's daughter. The Duke would have to explain that the Count's representative must have misunderstood something he said. Evidently the Duke is anxious to marry this particular girl, most likely because her father is rich. You would probably want to emphasize how much he admires the girl, how he would like to treat her with all the loving kindness she deserves--and you might suggest that his request for her dowry would be more modest than he had previously intended. It might be possible to ameliorate his bad original impression. The Count would probably like a marital alliance with the Duke and might be susceptible to persuasion. After all, he wasn't present but only has his agent's report to go by. It would be a hard letter to write, but there doesn't seem to be any other kind of letter you might be assigned to write on behalf of the Duke of Ferrara.