What mistake does Desdemona make when Othello asks for the handkerchief?

Although Shakespeare represents Desdemona as an
overwhelmingly virtuous character, he also shows us that she is a flawed human being
like everyone else, and can act unwisely.  This is true of her in two main contexts: her
impetuous elopement with Othello, revealed in Act 1, scene 1; and her dishonest response
to Othello's request to see the handkerchief in Act 3, scene
4. 

The truth of the matter is that Desdemona has
accidentally dropped the handkerchief in Act 3, scene 3 when mopping Othello's brow, in
perfectly innocent circumstances.  As usual, her motives are pure and, in this instance
a sign of love and concern.  In response to Othello's "I have a pain upon my forehead,
here" (3.3.288), Desdemona replies "Let me bind it hard..." (3.3.290).  Othello,
however, expresses an impatient desire to go inside and, following him, she drops the
handkerchief without noticing it fall to the ground.

By the
time the two meet again, Othello's mind has been so thoroughly poisoned by Iago, that he
is ready to attack Desdemona for the slightest sign of any indiscretion or unusual
behaviour.  Specifically, Iago has informed Othello in the interim that he has seen
Cassio wipe his beard (3.3.440-442) with a handkerchief “spotted with strawberries”
(3.3.438).  Othello sets Desdemona up by claiming to have a cold: “I have a salt and
sullen rheum offends me” (3.4.51).  He asks her to lend him her handkerchief, but,
unable to do so, she replies, “I have it not about me” (3.4.55).  Her main fault
consists in not explaining that she only very recently lost it which, of course, is
something that can happen to anyone.  We must remember that at one level, it is only a
handkerchief, but that, at another, it is to become for Othello the “ocular proof”
(3.3.365) of Desdemona’s infidelity.  Clearly, Desdemona knows that Othello is out of
sorts (“My Lord is not my Lord”: 3.4.125), but this is a very unwise first move. 
Othello is quick to pounce (‘That’s a fault.”: 3.4.57), and then proceeds to expound on
the magical properties of the handkerchief, and its significance as insurance against
infidelity. 

Desdemona then compounds her first mistake by
making three further inopportune comments.  The first of these is in response to
Othello’s revelation that the handkerchief was sewn by a prophetess from the silk of
sacred worms.  Her nervous and clumsy outburst, “Then would to God that I had never
seen’t!” (3.4.79) infuriates Othello and adds fuel to the fire of his anger.  The second
compounding fault is to lie to Othello by claiming that she will be able to produce the
handkerchief later: “Why so I can, sir: but I will not now” (3.4.82).  This is followed
immediately by her claiming light-heartedly that Othello’s over-reaction to the missing
handkerchief is a trick to prevent her from pleading Cassio’s case (3.4.89-90).  The
genius of Shakespeare allows the light-hearted response of Desdemona to become entangled
once again with Othello’s monstrous jealousy, and his now intense hatred of Cassio. 
Othello demands to have the handkerchief in three more rapid-fire outbursts, all of
which Desdemona meets with an innocent but misjudged reference to Cassio.  Othello exits
enraged.