What is the ruling on publishing a religious video clip that contains minor errors in recitation or pronunciation of Hadith, or citing a weak Hadith, or incorrect Sharia information, or a statement that has no precedent, even if the lecturer is trustworthy and most of his speech is beneficial?

Publishing beneficial da'wah lectures by trustworthy shaykhs is a good thing. The presence of minor errors in them does not prevent their publication, as such errors are slight and widespread. Preventing their publication would deprive people of much certain good for fear of a slight, potential harm. People do not intend to receive and memorize verses from the exact wording of the lecturing shaykh, so the pretext of people being affected by the shaykh's error in recitation is far-fetched. If it is possible to correct the errors through editing, then that is the most appropriate course of action; otherwise, they can be pointed out in writing.