Do the words in 33:34 authorise the sunna
of the prophet as a source of Islam?

And remember what is recited in your homes of God's revelations
and the wisdom. God is Gentle, All-Aware. 33:34

These glorious words in 33:34 are a continuation of the words in 33:33 which were addressed to the wives of prophet Muhammad. These words have often been misinterpreted to instate the hadith/sunna of the prophet as a lawful source of religion that must thus be followed by all believers.
The verse makes reference to:
1- God's revelations.
2- The 'hikmah' (wisdom).

The claim is that these words refer to two separate sources:
1- God's revelations refer to the Quran.
2- The 'hikmah' (wisdom) refer to the hadith/sunna of the prophet.

Once again, the core of this mistaken interpretation is the totally inaccurate interpretation of the word "hikmah" (wisdom). For the analysis, please refer to:
The word 'Hikmah' in the Quran

The claim is that the wives of the prophet were instructed to narrate to other people what they hear from the prophet (in their homes) of the Quran and also his hadith/sunnah. This in turn, they claim, gives legality to the hadith/sunna as a source of religion besides the Quran.

To further cement their understanding, we find numerous translations that took the liberty of adding a number of words (not found in the Arabic Quran) to their translation and interpretation of this verse. The following are some examples, with the added words marked in red:

Always˺ remember what is recited in your homes of Allah’s revelations and
˹prophetic˺ wisdom (which especially includes the sunnah of His messenger). Surely Allah is Most Subtle, All-Aware.(Translation by Mustafa Khattab)

Keep in mind and study what is recited in your homes of God’s revelations and the wisdom
(which especially includes the Sunnah of His Messenger). Surely God is All-Subtle (penetrating down into the innermost part of people’s heart and mind), All-Aware.(Translation by Ali Unal)

And remember what is recited in your houses, from Allah’s verses and wisdom; indeed Allah knows the minutest things, is All-Aware. (The Holy Qur’an and
the sayings- hadith - of the holy prophet.)(Translation by Faridul Haque)

And always remember the revelations of Allah and the
tradition and wisdom (of the holy prophet [blessings and peace be upon him]) which are recited in your homes. Surely, Allah is the Master of subtleties (for His favourite servants — Awliya’) and Well Aware (of all His Creation).(Translation by Mohammad Tahir-ul-Qadri)

And remember (O you the members of the prophet's family, the Graces of your Lord), that which is recited in your houses of the verses of Allah and al-hikmah (i.e.
Prophet's sunnah legal ways, etc. so give your thanks to Allah and glorify His praises for this Quran and the sunnah). Verily, Allah is ever Most-Courteous, Well-Acquainted with all things.(Translation by Muhsin Khan & Muhammad al-Hilali)

And remember the Verses of Allah and the wisdom
(prophet’s sayings) which are recited in your houses, surely, Allah is Courteous, Well-Acquainted.(Abdul Hye)

The reader is invited to compare the inaccurate translations above with the literal translation of 33:34 at the top of the page. Upon reading the correct Quranic words of 33:34, without the additions, we note the following:

1-
The word (prophetic) placed before (wisdom) is not in the Arabic text.

2-
The words (sunnah of His messenger) are not in the Arabic text.

3-
The word 'hikma' (wisdom) is repeated in the Quran as an attribute of the Quran "Al-Quran Al-Hakim" (36:2). This speaks of the Quran and the wisdom contained therein.

Also:

A.L.R. These are the signs of the 'Hakeem' (wise) Book. 10:1 (also 31:2 and 36:2)

4-
The claim that this verse instructs the wives of the prophet to tell others about what is recited in their homes is also without justification. The words in this verse specifically instructs the wives of the prophet to "remember" what they hear being recited in their homes. There is no mention of narrating to others.
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In addition to the above observations, God often gives us keywords in verses which God knows will be subject to dispute or manipulation. These keywords aid us in detecting the correct meaning and exposing the corrupt meaning.
When we look at the words in 33:34 we read the word 'yutla' which comes before

'ayat Allah wa al'hikmah'
yutla = being recited
Ayat Allah wa al-hikmah = God's revelations and the wisdom.

And remember what is 'yutla' (recited) in your homes of God's revelations and the wisdom. God is Gentle, All-Aware.
33:34

The word
'yutla' (verb) and Tilawa (noun), meaning (being recited) and (recitation) respectively, are always used in connection with the recitation of the verses of God's Book. The following are examples:

I have been commanded to be one of the submitters, and to
'atlu' (recite) the Quran." 27:91-92

T
hose who 'yatlun' (recite) God's Book, observe the Salat and give from what We have provided for them, secretly and publicly, are seeking a trade that can never go to waste. 35:29

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Normal conversation is spoken and not recited. It is God's Book that is recited. When people narrate human words of any person, they simply speak them, they do not recite them.
God was very careful to use the word 'yutla' (recited) in 33:34 to confirm that God is speaking about the recitation of the Quran. When God instructed the wives of the prophet to remember what is recited in their homes of God's revelations, God was speaking about the words of the Quran that are recited to them by the prophet and the wisdom contained in the Quranic words.