Assalamualaikum ustaz. Usually, the time between the two sermons is too short and brief. So, is it permissible for me to supplicate when the khatib is delivering the sermon? Hope for an explanation.
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
Among the advantages or favours of Fridays is that there is a time when supplications are granted. This is based on a hadith where the Prophet PBUH detailed the advantages of Fridays saying:
فِيها سَاعَةٌ لاَ يُوَافِقُهَا عَبْدٌ مُسْلِمٌ، وَهْوَ قَائِمٌ يُصَلِّى، يَسْأَلُ اللَّهَ تَعَالَى شَيْئًا إِلاَّ أَعْطَاهُ إِيَّاهُ
“There is a time in it (Friday) at which no Muslim would stand up while praying and beg Allah, Who is Great and Glorious, for something except that He would give it to him.” 
He (ﷺ) indicated with his hand that it (this time) is short.
In this issue, there are several opinions presented by scholars as mentioned by Imam al-Nawawi according to the opinion of al-Qadhi when he was commenting on this hadith in al-Minhaj Syarah Sahih Muslim, (6/140), among them are:
First: The time is when the khatib sits on the pulpit until after the prayer is completed. This opinion is supported by a hadith narrated by Abu Musa al-Asy’ari.
Second: The time is after asar until the sun sets.
Third: The time is when one is praying until he finishes his prayer.
Fourth: The time is when the imam exits until the prayer is completed.
Thus, the time for us to supplicate is open wide and the opinion that we’ve chosen is the second opinion according to the prophetic evidence.
Hence, the issue of supplicating when the khatib is delivering the sermon is impermissible. This is based on the hadith from Abu Hurairah RA, the Prophet PBUH said:
إِذَا قُلْتَ لِصَاحِبِكَ يَوْمَ الْجُمُعَةِ أَنْصِتْ. وَالإِمَامُ يَخْطُبُ فَقَدْ لَغَوْتَ
“When you tell your companion on Friday to be silent while the imam is preaching, you are guilty of idle talk.” 
Scholars have agreed that it is afdhal (prioritized) for the person attending the sermon from the imam to stay quiet and listen attentively. It is prioritized over him busying himself with dhikr alone, reciting al-Quran or supplicating. 
Saidina Ali RA narrated:
وَمَنْ لَغَا فَلَيْسَ لَهُ فِى جُمُعَتِهِ تِلْكَ شَىْءٌ
“Anyone who interrupts (during the sermon) will receive nothing (no reward) on that Friday.” 
We’d like to take this opportunity to suggest several additional sunnah acts that we can do on Fridays. We cite from al-Fiqh al-Manhaji, among them are:
First: It is sunnah to recite surah al-Kahfi on Friday night or its day
This is in accordance with a hadith from Abu Sa’id al-Khudri, the Prophet PBUH said:
مَنْ قَرَأَ سُوْرَةَ الكَهْفِ فِي يَوْمِ الجُمُعَةِ أَضَاءَ لَهُ مِنَ النُّوْرِ مَا بَيْنَ الجُمُعَتَيْنِ
“If anyone recites sūra al-Kahf (Qur’ān, 18) on Friday, light will shine brightly for him till the next Friday.” 
Second: Recite praise and salutations to the Prophet PBUH on Friday day and night
This is based on the hadith:
نَّ مِنْ أَفْضَلِ أَيَّامِكُمْ يَوْمَ الْجُمُعَةُِ فَأَكْثِرُوا عَلَىَّ مِنَ الصَّلاَةِ فِيهِ فَإِنَّ صَلاَتَكُمْ مَعْرُوضَةٌ عَلَىَّ
“Among the most excellent of your days is Friday; so, invoke more blessings on me that day, for your blessings will be submitted to me.” 
 Narrated by al-Bukhari (893) and Muslim (852)
 Narrated by al-Bukhari (892), Muslim (851) and others
 See Fath al-Bari, 8/274
 Narrated by Abu Daud (1051)
 Narrated by al-Nasaie (453)
 Narrated by Abu Daud (1047)