Is there any objection to occupying oneself with something else between the sunnah and fard of the obligatory prayers?

According to the preferred view in the Hanafi school, engaging in actions contrary to prayer—such as speaking without necessity, eating, or drinking—between the fard and sunnah of the obligatory prayers is considered mildly disliked (makruh tanzihan) because it reduces the reward of the prayer (Ibn Nujaym, al-Bahr, 2/53; Tahtawi, Hashiyah, 313). However, engaging in tasbih (glorification), dhikr (remembrance of Allah), or recitation of the Qur'an between the sunnah and fard, or the time spent going to the mosque if the sunnah is performed at home, is not regarded as an interruption. Therefore, there is no objection to these actions.

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu