Plot hole: Dr. Adam Soong is initially presented as a discredited scientist, banned from the scientific community; he gets debarred and his funding revoked. And it's not an internal matter; he is publicly exposed for it. His daughter in episode 6 even finds out this information on Google. Several news articles call him "mad scientist" and such. However, this same person at the same time throughout the rest of the season has every bit of pull and influence, not just through undercover channels, but is treated with the utmost honor and deference by the NASA PR people at public events.
Plot hole: Some part of the dialogue between Jean-Luc Picard and Admiral Kirsten Clancy must have been left on the editing room floor, because when Commodore Oh refers to the conversation the two had, says that "He referred to Zhat Vash by name." He did not. (00:21:30 - 00:37:30)
Plot hole: Mr. Vup is a Beta Annari, and they can, as it is stated (and for comedy purpose stated again) literally "smell" lies. However, Raffi gives Rios a unique concoction that camouflages lies, and it is made of drugs (beta blockers, anxiolytics, benzos). At least two things don't make sense here. First, Picard gets no shot and his whole flamboyant performance is one big lie from beginning to end, but he is not sniffed out - you'd also assume they could easily tell he has both eyes, since they have various detectors. Second, when the substances kick in as Rios is forced to lie openly, even us the audience, as olfactory-impaired as we are, can see he is getting high as a kite from him making a funny face; a species that can detect subtle changes in a metabolism over a simple lie, surely would detect when someone has such a dramatic alteration in front of their eyes - and see that as a telltale sign of something fishy going on.
Plot hole: In this episode, the 'Watcher' displays the power to possess/mind control people at will, jumping from body to body to set up the meeting with Picard. Forgetting the fact that this usage of power for such a menial task is actually detrimental to what she wants to do (it leaves more evidence, by unnecessarily messing up with minds just to tell the guy to go from A to B), this would have been super-useful for the rest of the season, but she just never ever uses it again, not even a nerfed version of it.
Plot hole: Adam Soong tosses the phaser before it explodes. The explosion is an underwhelming bang in the air that does not hurt anyone. Soong is a chubby old dude, with no access to teleport technology or any tech, no allies left in the area, inside Picard's home in France, and all he did was run out of the door, unarmed. All Rios (or anyone else, really, arguably even Picard-bot) has to do to catch him is comfortably jogged through the corridor, but somehow since he went out of the camera view and the plot doesn't want them to look for him, he ceased to exist, and he can stir trouble later after he gets (somehow) back to California.