When Othello spies Cassio with the handkerchief, what does he want to do?
Iago has Othello hide in Act IV, scene i, in order to watch him question Cassio about his whore. Othello believes that they are talking about Desdemona, while Cassio believes that the conversation is about Bianca, the harlot with whom he is having a relationship. The use of the pronoun "she" is most convenient here for Iago.
During the course of this, it is most fortuitous to Iago's scheme that Bianca comes storming in with Desdemona's handkerchief saying:
This is some minx's token, and I must take out the work? There! Give it your hobby-horse, wheresoever you had it...
And she apparently throws the handkerchief at Cassio.
Upon hearing this, Othello puts two and two together and arrives at the assumption that Cassio has received the handkerchief from Desdemona and given it to his whore. His response when Cassio leaves and he comes out of hiding is:
How shall I murder him, Iago?. . .I would have him nine years a-killing. . .And let her rot and perish and be damned to-night, for she shall not live. . .
He more than wants to kill both Cassio and Desdemona, he plans on it.