This hadith is narrated in Bukhari and Muslim:
قالت عائشة: لما ثقل النبي صلى الله عليه وسلم، واشتد وجعه استأذن أزواجه أن يمرض في بيتي، فأذن له، فخرج بين رجلين تخط رجلاه الأرض، وكان بين العباس ورجل آخر قال عبيد الله: فذكرت ذلك لابن عباس ما قالت عائشة، فقال لي: وهل تدري من الرجل الذي لم تسم عائشة؟ قلت: لا، قال: هو علي بن أبي طالب
Narrated from ... from Ma'amar from al-Zuhri from Ubaidullah bin Abdullah from Ayesha:
When the Prophet (ﷺ) became seriously ill and his disease became aggravated he asked for permission from his wives to be nursed in my house and he was allowed. He came out with the help of two men and his legs were dragging on the ground. He was between Al-Abbas and another man."
Ubaidullah said, I told Ibn Abbas what Aisha had narrated and he said, 'Do you know who was the (second) man whose name Aisha did not mention'
I said, 'No.'
Ibn Abbas said, 'He was Ali Ibn Abi Talib.'
— Bukhari and Muslim
However other collections record an addition at the end such as:
ولكن عائشة لا تطيب لها نفسا بخير
Which means that Ayesha was displeased with Ali and did not not make any complimentary mention of him. It is this addition that is underlined in your image.
This is speculation by one of the narrators of the hadith, or maybe even by Ibn Abbas. However it is only speculation and there are other possible explanations, e.g. she could have forgotten or have been unable to see him clearly, or could have omitted the name because some traditions imply that the second person changed.
As for the assumption that Ayesha disliked mentioning the name of Ali, then this is unlikely or at best may have been a temporary quarrel, as there are ahadith in which she mentions him in favorable terms, such as:
أتيت عائشة أسألها عن المسح على الخفين، فقالت: عليك بابن أبي طالب، فسله فإنه كان يسافر مع رسول الله صلى الله عليه وسلم
I came to A'isha and asked her about wiping over the socks. She said: "You better go to Ibn Abi Talib for he used to travel with Allah's Messenger (ﷺ) ... "
ائت عليا فإنه أعلم بذلك مني
Go to Ali, for he knows more about that than I do.
— Muslim
قالت عائشة: خرج النبي صلى الله عليه وسلم غداة وعليه مرط مرحل ... ثم جاء علي فأدخله، ثم قال: {إنما يريد الله ليذهب عنكم الرجس أهل البيت ويطهركم تطهيرا} [الأحزاب: 33]
Aisha said: The Prophet ﷺ went out one morning wearing a striped cloak of black camel hair ... then Ali came and he enfolded him in it then said: "Allah intends only to remove from you the impurity [of sin], O people of the [Prophet's] household, and to purify you with [extensive] purification." (33:33)
— Muslim