Hamza Konuk, in his thesis, VÂNÎ MEHMET EFENDİ'NİN MÜNŞE'ÂTI (TRANSKRİPSİYON, TAHLİL VE DEĞERLENDİRME) and Ayşegül Ünal Akkaya, in her 2019 thesis, (Köprülü Ayşe Hanım ve Osmanlı'da Hâne Politikaları) mentioned a letter from Fâzıl Ahmed Paşa, the Grand Vizier from 1661 to 1676, to Murad IV's wife Ayşe Haseki.
This is the original letter:
“Âyişe
Sultân Hazretlerine Yazılmışdır:
Devletlü ve sa'âdetlü sultânım hazretlerinin hûk-i pâ-yı şerîflerine kemâl-i ta'zîm ü ikrâm ile rûy-mâl ve devâm-ı ömr ü devletleri du'âsı tekrâr u tezkâr kılındıkdan sonra ma'rûz-u bende-i sadâkat-kârları budur ki,
eymen-i evkâtda mektûb-u şerîfleri gelüb mefhûmunda her ne ki işâret buyurulmuş ise ma'lûm-u bendeleri oldu. Nevâziş ü ihsânlarıyla bendelerin teşrîf ü tekrîm buyurub ögmüşler mekârim ü eltâf-ı vâfirelerinden mahzûz u memnûn olmuşuzdur. Hakk Sübhânehû ve Ta'âlâ Hazretleri tûl-ü ömr ile mu'ammer idüb hemîşe safâ vü sıhhat ve selâmet ü 'âfiyet ile eyyâm-ı ömr ü devletlerin ber-devâm eyleye.
'İzzetlü paşa birâderim hazretlerîçün işâret buyurulmuş, in şâa'llâhu Ta'âlâ vaktiyle mesrûr u ber-murâd olmalarına sa'y ü takayyüd ideriz. Rızâ-yı şerîferi üzere harekete sa'y ider kullarıyız. Hemân sa'âdetlü sultânımın du'â-yı hayırları recâ vü niyâz olunur.
Bâkî lütf u ihsân sa'âdetlü sultânım hazretlerinindir.”
In English:
To the blessed feet of my noble and fortunate Sultan, I humbly present myself with the utmost reverence and respect, and, after repeatedly praying for the continuation of Your life and prosperity, I submit that:
Your blessed letter arrived at an auspicious time, and I have understood all that You indicated therein. You have honored and favored Your servants with Your kindness and generosity, and, by praising us, You have made us exceedingly pleased and grateful for Your abundant grace and benevolence. May Almighty God grant You a long life and always preserve You in joy, health, peace, and well-being.
You have also indicated your wishes concerning my honored brother, the Pasha. God willing, we shall endeavor in due course to ensure that he is pleased and obtains what he desires.
We are but Your servants who strive to act according to Your blessed will. We ask and earnestly pray only for the good prayers of my fortunate Sultan.
The rest of the favor and generosity belong to my fortunate Sultan.
Now, why would the Grand Vizier send letters to the haseki of a dead sultan ? Why would "he strive to act according to her will" ? Murad was already gone for him to care about her favor.. Well, God knows ...
Also, According to Von Hammer and Venetian reports: In 1631, Ayşe Haseki gave birth to a son and was being approached/appealed to by the grandees, especially by the Kapitanbaşı, whose restoration to the position that he lost was anticipated by everyone because he was in the good graces of the Haseki. However, before that could happen, when he was sent to Rumelia to assemble the troops, he died after being poisoned by his brother-in-law, Rejeb.
In the same Venetian report, it is mentioned that Asan Pasha, through Kösem, attempted to have himself elected Caïmacam, but the plot was discovered. Mehmet Ağa wanted to become Defterdar; he lost his head. (Vol 9, p 163, Von Hammer, Histoire de l'Empire ottoman)
So, Ayşe had some sort of political connections, and everyone knew about the Kapitanbaşı being in her good graces and that his restoration was a matter of time. Alessandrini also mentioned that she was ambitious, in addition to Foscarini mentioning that she was involved in the princes’ execution, etc. So, either Murad was not aware of this, or he was aware and let it pass since she was not a threat (her power came from him and his love for her, after all), because his mother was punished when she did things like this. I’m pretty sure Kösem was exiled a few times during Murad’s reign for trying to do anything related to politics.