Praise be to Allah.
There is nothing wrong with the khateeb saying at the end of his speech, “Aqoolu haadha wa astaghfir-Allaah (I say this and I ask Allah for forgiveness)” or “Aqoolu qawli haadha wa astaghfir-Allaah (I say these words of mine and I ask Allah for forgiveness)”, whether this is the first or second Friday khutbah or any other kind of speech. These are words that are narrated from the Prophet (blessings and peace of Allah be upon him) and from the khateebs among his Companions (may Allah be pleased with them).
Ibn Hibbaan (3828) narrated from Ibn ‘Umar (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) gave a speech on the day of the conquest of Makkah. He praised and glorified Allah then he said: “To proceed: O people, Allah has taken away from you the pride in ancestry of Jaahiliyyah. O people, people are of two types, the righteous and pious one who is dear to his Lord and the wretched evildoer who is worthless before his Lord.” Then he recited the verse (interpretation of the meaning): “O mankind! We have created you from a male and a female, and made you into nations and tribes, that you may know one another. Verily, the most honourable of you with Allaah is that (believer) who has At‑Taqwa [i.e. he is one of the Muttaqoon (the pious. See V.2:2)]. Verily, Allaah is All‑Knowing, All‑Aware” [al-Hujuraat 49:13]. Then he said: “Aqoolu haadha wa astaghfir-Allaah (I say this and I ask Allah for forgiveness).”
This hadeeth was classed as saheeh by Shu’ayb al-Arna’oot in his Tahqeeq, and by al-Albaani in al-Silsilah al-Saheehah, 2803. The biographers narrated that Thaabit ibn Qays ibn Shammaas, the khateeb of the Ansaar (may Allah be pleased with him) said it in the presence of the Prophet (blessings and peace of Allah be upon him). Ibn Ishaaq said in the report of the delegation of Banu Tameem and their boasting in front of the Prophet (blessings and peace of Allah upon him): The Messenger of Allah (blessings and peace of Allah upon him) said to Thaabit ibn Qays ibn al-Shammaas, the brother of Banu al-Haarith ibn al-Khazraj: Get up and answer the man’s speech. Thaabit got up and said: Praise be to Allah Who created the heavens and the earth and established His rule therein, and His knowledge encompasses His Throne; nothing exists but by His bounty. By His power He made us kings and chose the best of His creation as a Messenger, who is the most honourable in lineage, the most sincere in speech, and the noblest in ancestry, and sent down to him His Book and entrusted him (to guide) His creation. He was Allah's choice from among the worlds. Then He summoned men to believe in him, and the emigrants (Muhaajireen) from his people and his kinsmen believed in Allah's Messenger; the most honourable of men in lineage, the highest in dignity, and the best in deeds. The first of creation to answer and respond to Allah when the Messenger called them were ourselves. We are Allah's helpers and the assistants of His Messenger, and we will fight men until they believe in Allah. He who believes in Allah and His Messenger has protected his life and property from us; and he who disbelieves we will fight for the sake of Allah unceasingly and killing him will be a small matter to us. These are my words and I seek Allah's forgiveness for myself and the believers, both men and women. Peace be upon you.
End quote from al-Seerah al-Nabawiyyah by Ibn Hishaam, 2/562; al-Seerah by Ibn Katheer, 4/79.
These words were also narrated from Abu Bakr, ‘Umar and ‘Ali (may Allah be pleased with them) in their khutbahs, and some contemporary scholars use them in their khutbahs such as Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him).
As for the objection to the khateeb mentioning things that are not his own words, such as verses, hadeeths and reports, this objection is not valid, because what these words refer to is everything that the khateeb said in his khutbah, even the verses, hadeeths and reports that the khateeb quoted.
The point is that saying these words is permissible and there is nothing wrong with it, and there is no justification for objecting to it.
We ask Allah to help us to understand His religion properly.
And Allah knows best.