This Kathīr is Ibn Qulayb ibn Mawhab al-Baṣrī. A distinction has been made between him and Kathīr ibn Murrah ibn Yūnus, and al-Ḥāfiẓ followed this view, describing the former as trustworthy and the latter as acceptable.
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Ibn Shāhīn recorded in his work on the Companions, through a weak route from Abān ibn Abī ʿAyyāsh—one of the narrators deemed abandoned—from Anas that Abū Fāṭimah al-Anṣārī came to the Messenger of Allah, peace and blessings be upon him, and then mentioned the third passage, concerning fasting. This is how it appears in Ibn Ḥajar’s al-Iṣābah. Yet this passage has an authentic corroborating report from Abū Umāmah, attributed to the Prophet, with similar wording. Al-Nasāʾī recorded it, and Ibn Khuzaymah, Ibn Ḥibbān, and al-Ḥākim authenticated it. It is referenced in Takhrīj al-Targhīb (2/61–62). Al-Ṭabarānī also recorded it (7463–7465), with an authentic chain of transmission.
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Ibn Mājah recorded (1/435), through the route of al-Walīd ibn Muslim: ʿAbd al-Raḥmān ibn Thābit ibn Thawbān related to us, from his father, from Makhūl, from Kathīr ibn Murrah, that Abū Fāṭimah related to him, saying: “I said: ‘O Messenger of Allah, tell me of a deed to which I may remain steadfast and perform.’ He said: ‘Adhere to prostration …’” And so on.
I say: This is a sound chain of transmission, as al-Mundhirī said (1/145). Aḥmad (3/428) and al-Dūlābī, in al-Kunā (1/48), also recorded it through the route of Ibn Lahīʿah, who said: Al-Ḥārith ibn Yazīd related to us, from Kathīr al-Aʿraj al-Ṣadafī, who said: “I heard Abū Fāṭimah, while he was with us at Dhū al-Fawārī, saying …” He then cited it as a report attributed to the Prophet, with the wording: “O Abū Fāṭimah, increase your prostration …” This hadith does not include the words: “and erase from you one sin.”
I say: Its narrators are trustworthy apart from Kathīr, who is Ibn Qulayb. Al-Dhahabī said: “An Egyptian narrator who is unknown.” It has also been said that he is Kathīr ibn Murrah mentioned in the preceding route. And Allah knows best.