Why does Desdemona push so hard for Cassio to get his job back?
Iago has convinced Cassio to appeal to Desdemona to intervene on his behalf with Othello. Cassio was fired because of his behavior (Iago gets him drunk, and then Cassio reacts violently when he is insulted). Othello fires Cassio, who feels terrible about what has happened. Cassio is friendly with Desdemona, and Iago convinces Cassio that he should ask Desdemona to ask Othello, who is after all her husband, to reinstate Cassio. Desdemona seems to ask this favor of Othello in a sort of "if you love me you will do this" sort of way. Othello's reluctance irritates her, and she keeps asking him to reinstate Cassio. For the play's plot, Desdemona has to take Cassio's part with passion, because Iago uses her defense of Cassio to frame them in their "affair."
Desdemona has promised to help Cassio get his job back. As the woman that she is, she will not go back on her word. Shakespear may have added this in to fuel Othello's anger towards Cassio. He is already upset with Cassio for causing such a scene so by Desdemona constantly bagering him about Cassio he would most likey get even more mad at Cassio. She is fueling the fire so to say!