When Picard helps himself to Riker's bottle of wine, he picks it up with his right hand and has his left at his side. Then the shot changes and now his left hand is in his pocket.
The on-screen text that accompanies the opening flashback states that it happened 3 weeks prior to the "present day" of the story. However, in The Impossible Box (2020), which took place just moments before Nepenthe, Hugh stated that Narek had arrived on the Artifact 2 weeks prior to the "present day" of the story. The events of the flashback happened after Narek arrived on the Artifact.
At 15:14, there's not a mark on Soji's face, but, 10 seconds later ,she has re-acquired the bloodstains she got while escaping from the Romulan Meditation chamber.
In most of the exterior shots of Riker's pizza oven, there is no smoke coming from the chimney despite having a fire within. However, since smoke is one of the worst aspects of barbecues, 24th century technology may have developed a device to eliminate it.
Narek's snakehead ship is said to be staying just out of sensor range of La Sirena, yet exterior shots show them very close to one another, and La Sirena is clearly visible to the naked eye from the snakehead's cockpit. The two ships are, are best, a couple hundred meters from one another. Starship sensors have a far greater range than human (or Romulan) eyeballs, extending many light years. The snakehead ship would be well within La Sirena's sensor range.
Kestra self-identifies as a pacifist, and Star Trek: The Next Generation (1987) established that killing animals for food was long out of fashion in the Federation, yet Kestra kills a bunnycorn and gives the carcass to her father to make sausage for the pizza.
Riker mentions "Newton's fourth law of thermodynamics" - unlike Newton's laws of motion, the laws of thermodynamics were not compiled by Isaac Newton, nor by any single individual, but by consensus. So it is incorrect to apply Newton's name to all or any of the laws of thermodynamics, even a humorous one.