off the cuff

off the cuff (spontaneous, without planning) — без подготовки, экспромтом; на ходу (=по ходу событий, без подготовки)

So, if you speak off the cuff, you do it without having planned what you will say, without having prepared or thought about your words first

to play it off the cuff — действовать по обстоятельствам

An off-the-cuff remark is one that is not planned (without previous thought or preparation)

Example 1: She hadn't prepared a speech and just said a few things off the cuff. — Она не готовилась к выступлению заранее и просто сказала несколько слов экспромтом.

Example 2: I'm just speaking off the cuff here — I haven't seen the results yet. — Я сейчас импровизирую, говорю без подготовки — результатов я еще не видел.

Example 3: Brian's decision to propose was completely off the cuff. He didn't even have a ring that night.

Related vocabulary (speaking off the cuff):
to ad-lib (sth) (to improvise speech; speak or perform in public without previously preparing one’s words)
To ad-lib means to do something "off the cuff" — spontaneously, with no planning in advance. With actors forgetting their lies, this means they would simply say something, — anything, until they get back on track.
an ad-lib (remark made spontaneously without prior preparation)

Example 1: The actor had to ad lib because he’d forgotten his script/his lines.

Example 2: She ad libbed half the speech.

Example 3: His ad-libs got him in trouble with the politicians.

speak impromptu,
speak extemporaneously,
invent, make up,

[play it by ear]
[wing it]
[с ходу]