If al-Hakim says in al-Mustadrak (this hadith is on the condition of al-Bukhari and he did not include it) and al-Dhahabi agrees with him on that

The questioner : If al-Hakim says in his al-Mustadrak: this hadith is on the condition of al-Bukhari and he did not include it, and al-Dhahabi approves of that, or if al-Dhahabi says about the hadith that it is authentic, is such a hadith relied upon in that case? And is it cited as evidence in the same way any hadith in the two Sahihs is cited?

The Shaykh : Here, the answer differs for the general public and for some of the specialists. By these specialists, I mean the scholars of hadith. As for the general public, their obligation, as we began in the word of this session, is to ask the people of knowledge. Undoubtedly, al-Hakim is among the people of knowledge of hadith in terms of authenticating and weakening, especially if we compare him to those who have no knowledge. I do not mean the general public, but rather the specialists of the last times who do not study the science of hadith. It is obligatory for them to refer to him and look at his position regarding the hadith. If he authenticated it, they should follow him, unless it becomes clear that al-Hakim made a mistake according to the judgment of someone else who is more knowledgeable about the science of hadith than he is. All the more so, if al-Hakim authenticated a hadith and al-Dhahabi agreed with him, the general Muslims should follow that, unless it becomes clear to them through a report from a scholar that al-Hakim was in error and al-Dhahabi also followed him in his error. In that case, they should return from following him or both of them to following the correct position that became clear to them from others. As for the specialists among the scholars of hadith, due to the knowledge they have acquired regarding the biographies of the narrators of hadith first, and the knowledge they have acquired regarding the terminology of hadith second, it is not obligatory for them to follow al-Hakim, nor even al-Dhahabi, because it becomes clear to them in many cases that much error and mistake occurred in their authentication. In short, just as it is obligatory for the general Muslims...
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In one case, when it becomes clear to them that that jurist or that hadith scholar made a mistake, it is not permissible to follow the mistake.