Learn Italian: Avendolo ripetuto, l'attore ha migliorato il gesto. - Having repeated it, the actor improved the gesture.

In past gerunds, pronouns must attach to the end of the auxiliary verb (avendo or essendo), not the past participle. Therefore, we say avendolo ripetuto, never avendo ripetutolo.

Does the past participle ripetuto have to agree with the attached pronoun lo?

Yes. Just like in the passato prossimo, when you attach a third-person direct object pronoun (lo, la, li, le) to the auxiliary verb, the past participle must agree with it. Here, lo refers to il gesto (masculine singular), so ripetuto ends in -o. If the actor had repeated a feminine noun like la scena, it would be avendola ripetuta.

Can I use this gerund form if someone else repeated the gesture, but the actor improved it?

No. A standalone gerund clause like avendolo ripetuto almost always shares its subject with the main clause. The sentence tells us that the actor is both the one who repeated the gesture and the one who improved it. If the subjects were different, you would need a full clause, such as Dato che il regista lo aveva ripetuto... (Since the director had repeated it...).

Is there another common way to express Avendolo ripetuto?