Plot hole: The plot resolution hinges on the fact that the culprit gave the illusion of the door being locked while the lock was already broken. And he did that by jamming under the door a rusty old fork, through the usage of a string. However, this appears really far-fetched to say the least; if the door was obstructed by an object under it, it would have not given way as if the lock was busted. It would have dragged, created a screeching noise and scraping the floor leaving visible marks (which considered it's a locked room mystery, would have been investigated). The person breaking in was Jack, even, who is supposed to be really perceptive and crime-savvy and not the average person.
Plot hole: The way Jack narrates the murder does not make sense; according to him, the killer went first to grab the telephone, then back to the garden to kill the victim, then from there, he had to go, unnoticed and with his robe loose, back to the shack (which seems to be close to the entrance and the box where the phone was). Moreover, to stick the rope into the clay sculpture, especially the way we see it, not entwined into a ball but unfurled, he needed to disrupt the sculpture in a way that would have been noticeable, even if the clay was not entirely dry yet.
Plot hole: One of the guests is a journalist going undercover. The police finds him out because...they google his name, "Bryn Williams journalist" (no quotation marks), but then not finding anything (and Jack says so despite actual results being visualized, but he dismisses them at a glance) Florence has a stroke of genius and says "William Bryn then." And this time the googling pays off, with a search result page that says, literally, that he's an investigative journalist "known for going undercover to investigate", apparently being a master at that having won prizes! In all this amazing silliness, it appears quite impossible that they wouldn't know his real name, since they already identified the suspects and ran background checks, which in every episode always include checking with immigration when they entered Saint Marie. He couldn't have entered the country under a fake name.
Plot hole: The murder as described makes kinda sense (it's a device used in other works of fiction), but not at all as shown; the person hiding in the bathroom closes the door to the point of making it click, and opens it also with a click. Even panicked, the other person in the room is going to hear the sound of a lock right behind him. The door should have been closed only partially. Moreover, the bathroom door is right by the front door closed just by the door chain, and Philip Marston had the door unlocked and with just the chain to break; the bathroom door was fully in his view the whole time.
Plot hole: There is simply no way that the cheating presented in the episode, with the player reaching in his pocket to mark the cards, consistently for over 5 years (!), and wearing the same obvious prop, could go unnoticed, especially in official tournaments with millionaire prizes, on camera even.
The Healer - S7-E4
Plot hole: If the person pretending to be Steadman wrote to his fiancée breaking off the engagement, wouldn't she recognise that it wasn't his handwriting? They had known each other for years, surely she had seen his handwriting before.
Plot hole: The author uses a typewriter and avoids technology. His fake identity was built on a laptop kept on another island. But in the big reveal, her guilt is predicated on her having seen the laptop to find out he was leaving her.
Plot hole: At the beginning of the episode it is established that Samuel Palmer "was putting out his rubbish just after midnight", but in the rest of the case (coroner report, case discussion at the station, question to the suspects) the time of death is 10 PM. The part of the dialogue at the beginning (Dwayne talking to Jack and Florence outside the house) is cut in home video releases, but still appears in Jack's case-solving flashback reel. Other references to the wrong midnight time appear in Cordell' statement as he turns himself in (says he entered from the window " Just before midnight.") and Jack's usual schtick with the suspects ("Shortly after midnight, Eugene's neighbour, Samuel, he heard arguing").
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.