Al-Hakim and a-Dhahabi disagree on the qualification of the hadith
It must be noted that the hadith reveals a view held by sufis which can be summarized as saying that if Allah didn't create Muhammad() HE wouldn't have created any of the creations.
First of all if you check in al-Hakim's al-Mustadrak you may find below or after this report his comment giving his qualification:
In the following I'll translate from Arabic language, as these translation is of my own take it carefully
هذا حديث صحيح الإسناد ولم يخرجاه
This hadith has a sound isnaad but they (both: referring to al-Bukhari and Muslim) didn't compile it.
The link for the Arabic original also provides imam a-Dhahabi's comment on the specific hadith from at-Talkhess (a summary of al-Mustdarak compiled by imam al-Bukhari in his youth) which shows his qualification:
أظنه موضوعا على سعيد
I suppose it was fabricated and (falsely) attributed to Sa'id (ibn al-Mussayib)
Another similar report from al-Mustradrak and it's authenticity
Before referring to other qualifications there's also another report in al-Mustadrak with the same meaning on the authorithy of 'umar ibn al-Khattab, which got the same qualifications of both above scholars:
عن عمر بن الخطاب رضي الله عنه، قال: قال رسول الله صلى الله عليه وسلم: " لما اقترف آدم الخطيئة قال: يا رب أسألك بحق محمد لما غفرت لي، فقال الله: يا آدم، وكيف عرفت محمدا ولم أخلقه؟ قال: يا رب، لأنك لما خلقتني بيدك ونفخت في من روحك رفعت رأسي فرأيت على قوائم العرش مكتوبا لا إله إلا الله محمد رسول الله فعلمت أنك لم تضف إلى اسمك إلا أحب الخلق إليك، فقال الله: صدقت يا آدم، إنه لأحب الخلق إلي ادعني بحقه فقد غفرت لك ولولا محمد ما خلقتك (Source)
Omar ibn al-Khattab, may Allah be pleased with him, said: The Messenger of Allah, may Allah’s prayers and blessings be upon him, said: “When Adam committed a sin, he said: O Lord, I ask you for the sake of Muhammad, why did you forgive me?” Allah said: “O Adam, how did you know Muhammad when I did not create him?” He said: “O Lord.” Because when you created me with your hand and breathed into me from your spirit, I raised my head and saw on the pillars of the throne written: There is no god but Allah and Muhammad is the Messenger of Allah, so I knew that you did not add anything to your name except the most beloved of creation to you. And had it not been for Muhammad, I would not have created you
The qualification of this hadith was discussed in fatwa islamqa #128173. This hadith was also qualified as fabricated by ibn Taymiyyah. However among the scholars who disagreed is al-Hafidh Taqy a-Din as-Subki in his Shifa' as-Siqaam fi zyarati khayri al-Anaam شفاء السقام في زيارة خير الأنام. in which he assumed if ibn Taymiyyah would have been aware of our hadith in question which was compiled by al-Hakim he wouldn't have qualified this hadith in such a manner. It must be noted that beside his great knowledge in almost all disciplines of islamic knowledge he was a known sufi, therefore you may rather find his book on shi'a websites or only in pdf-form than on sunni websites (my source), as the later mostly propagate salafi approved books. Here a part of the quote from Shif'a as-Siqam with a translation of my own with some support from google:
والحديث المذكور لم يقف عليه ابن تيمية بهذا الإسناد، ولأبلغه أن الحاكم صححه.
فإنه قال - أعني ابن تيمية -: (أما ما ذكره في قصة آدم من توسله، فليس له أصل، ولا نقله أحد عن النبي صلى الله عليه وآله وسلم بإسناد يصلح الاعتماد عليه، ولا الاعتبار، ولا الاستشهاد).
ثم ادعى ابن تيمية أنه كذب، وأطال الكلام في ذلك جدا بما لا حاصل تحته، بالوهم والتخرص، ولو بلغه أن الحاكم صححه لما قال ذلك، أو لتعرض للجواب عنه.
The aforementioned hadith was not found by ibn Taymiyyah with this chain of transmission, and nor was he informed that al-Hakim authenticated it.
For he said - I mean Ibn Taymiyyah -: (As for what he mentioned in the story of Adam of his supplication, it has no basis, and no one transmitted it from the Prophet, may God’s prayers and peace be upon him and his family, with a chain of narrators that can be relied upon, nor considered, nor cited).
Then Ibn Taymiyyah claimed that he had lied, and he spoke at length about that very much, with what is not possible under him, by delusion and deception, and if he had reached him that al-Hakim corrected him, he would not have said that, or he would have offered the answer to him.
It must be noted here that it is doubtful that as-Subki correctly quoted ibn Taymiyyah because he mentioned the hadith and the issue with 'Abdarrahman ibn Zaid ibn Aslam عبد الرحمن بن زيد بن أسلم who was known for his mistakes as summarized in the fatwa mentioned above (see the hadith mentioned after #493 here howver the discussion goes over several pages in the same book). The wording reported by as-Subki neither seem to appear in ibn Taymiyyah's book about tawasul (linked above) nor in his discussion of the topic in his Majmu' al-Fatawa. As matter of fact ibn Taymiyyah pointed at the fact that al-Hakim himself declared 'Abdarrahman ibn Zaid ibn Aslam as da'if in his book "al-Madkhal ila ma'arifati as-Sahih mina as-Saqeem المدخل إلى معرفة الصحيح من السقيم":
٤٩٥ - قلت: ورواية الحاكم لهذا الحديث مما أنكر عليه، فإنه نفسه قد قال في كتاب المدخل إلى معرفة الصحيح من السقيم: عبد الرحمن بن زيد بن أسلم روى عن أبيه أحاديث موضوعة لا يخفى على من تأملها من أهل الصنعة أن الحمل فيها عليه. (Quote from ibn Taymiyyah's book)
495 - I said: Al-Hakim’s narration of this hadith is something he denounced, for he himself said in the book “The Introduction to Knowledge of the Sahih(sane) from the Saqeem (ill)”: Abd al-Rahman bin Zayd bin Aslam narrated fabricated hadiths on the authority of his father, and it is not hidden from those who contemplate them from the people of the profession that the burden in them is upon him.
Some more qualification of the hadith
As for the hadith in question we may find a fatwa islamqa #23290 in which it was mentioned -according this fatwa- both al-Albani and ibn Hajar agreed with a-Dhahabi's verdict on this hadith. Here the quote from this fatwa:
Al-Hakim said: its isnad (chain of narration) is saheeh! But al-Thahabi commented on this and said: I believe it is fabricated and falsely attributed to Sa’eed.
Meaning, Sa’eed ibn Abu ‘Aroobah (one of the narrators of this hadeeth). This hadeeth was narrated from him by ‘Amr ibn Aws al-Ansari, who is the one who is accused of fabricating it. Al-Thahabi mentioned him in al-Meezan where he said: “He produced a munkar (rejected) report,” then he quoted this hadeeth, and said, “I believe that it is mawdoo’ (fabricated).” Al-Hafiz Ibn Hajar agreed with him, as it says in al-Lisan.
Al-Albani said in al-Silsilah al-Da’eefah (280): There is no basis for it.
