Should one who is reciting the morning remembrances (adhkar) pause upon hearing the rooster crowing to ask Allah for His bounty, even if the remembrance they are saying does not include asking for bounty? And should the remembrance after prayer or the recitation of the Qur'an be interrupted if a Muslim speaks for some reason, and what is the criterion for that?

It is legislated to supplicate and ask Allah for His bounty upon hearing the crowing of roosters, and it is permissible to interrupt recitation or ذكر (remembrance of Allah) for that purpose. If a Muslim interrupts his recitation or ذكر for a reason such as returning a greeting, answering a mother's call, or enjoining good and forbidding evil, there is no harm in it. He should then resume his recitation or ذكر after the reason has passed, because ذكر and recitation are acts of worship, and what has already been done of them is not invalidated by interrupting them.