He asks Desdemona, but it is at Iago's insistence that he does this. In Act II, scene iii, Iago encourages Cassio to "mend [the loss of his position] for your own good." And, even though Cassio believes that asking is a lost cause because Othello will "tell me I am a drunkard," Iago tells him not to give up hope. He persuades Cassio to ask Desdemona for help getting his place back:
Our general's wife is now the general . . . . Confess yourself freely to her, importune her, she'll help to put you in your place again. . . This brawl between you and her husband, entreat her to splinter and. . .this crack of your love shall grow stronger than 'twas before.
Pretty sound advice, right? But once Cassio exits, Iago shares his real motives to the audience:
. . . Divinity of hell!
When devils will their blackest sins put on,
They do suggest at first with heavenly shows,
As I do now. For while this honest fool
Plies Desdemona to repair his fortunes,
And she for him pleads strongly to the Moor,
I'll pour this pestilence into his ear,
That she repeals him for her body's lust. . .
And so, the real turning point of Iago's scheme has arrived -- the opportunity to convince Othello that his wife is having an affair with Cassio.