Plot hole: When we see the murder happen in flashback, the killer stabs the victim wearing no gloves whatsoever, nor wipes the handle. In a very sportsman conduct, the victim also decides not to literally scream bloody murder as he gets stabbed, nor make any sort of sudden, noise inducing movement that would have instantly exposed what was happening. He gives their killer, apparently, all the time to go back to their accomplice before Catherine notices the blood pouring on the floor - how rude of him to silently bleed all over the booth without cleaning after himself. (00:41:45)
Plot hole: The murder happens where and when it happens because the candidate "is a very busy man", and apparently then the best course of action to kill him is doing it while he is casting his vote. At this operation, involving the other 2 candidates for the role, there is no press nor any normal voter, for no reason - not safety since Dwayne was not expected. Had they introduced the rich Victor Pearce as some sort of mobster surrounded by bodyguards, it would have been an acceptable plot idea, but the guy travels with his son as sole member of the staff and nothing about his characterization leads the viewer to believe that the only chance to murder him is for a rotund 62 years old lady in clogs to perform a Metal Gear stunt sneaking in undetected while a priest is facing the other way for 5 seconds and pray that nobody else shows up at the voting booth and all the others are taking their time to put a cross on a piece of paper.
In the Footsteps of a Killer - S6-E7
Plot hole: Somehow, Jack is so lax in his investigation that he does not ask any detail about the sales representative supposedly the husband had an affair with and that he ended (which could have very well been a suspect nobody considered before, since he ended the relationship to stay with his wife...had she existed, but Jack can't know that he's lying!), but has acquired DNA from Ian Matlock to run an overnight test on the hair sample (which we have to assume was complete with follicles and still in test shape after 8 years in a bag).
In the Footsteps of a Killer - S6-E7
Plot hole: Spoiler. The killer needs to get rid of two women. One is his mistress that he just dumped and is acting psychotic about it, the other is his wife. He kills the wife and makes it look like it was his mistress to murder her. So she goes to jail, where she will die two years later due to pneumonia...and in all this time, she never once says anything about the affair! He has been extraordinarily lucky, because had she said anything, and she had absolutely no reason not to in her circumstances, the case would have not been so open-and-close, they would have considered the idea that he could have been an accomplice, but even assuming the past detective (which DI Richard Poole called a good detective) was a total fool, at the very least Jack and his team would have found a trace of this controversial alleged affair in the files and solved the case much earlier. But no, the plan was to send his mistress to jail and that somehow made her cease to be a threat to him, when instead she'd have been much more dangerous to him.
In the Footsteps of a Killer - S6-E7
Plot hole: The original murder went unnoticed because of extraordinary incompetence of the police; even if they did not have the ingenuity to extrapolate the background noise, they still had to investigate how a murderer would have gotten away with a corpse in the middle of a carnival, but the issue is never raised. More importantly, the last phone call of the victim came from a place 40 minutes away from the city, which would mean a different cell tower - and since they needed to track down the body that never turned out, monitoring the phone the last call came from is standard procedure.
Plot hole: The big climax of the episode involves Humphrey arriving at 1 PM to escort Martha to the airport as he promised, finding the shack empty with a parting letter from her and rushing on Dwayne's motorbike to catch her plane, failing to do so. All very cinematic, but that means that had Martha not left early, she could have not caught the plane at all.
The Secret of the Flame Tree - S6-E2
Plot hole: For the murder to happen as described, the victim must have set an appointment at the cliff with a mentally ill, barely coherent person who spends her life holed up in a bungalow and is afraid of everyone, trusting her to remember and be on time - instead of knocking at her door and walk her to the location, that is very close. It's a complete absurdity.
Plot hole: The big locked room mystery is such only because the detectives build it up as one; they don't make any remark about the fact that the door can be locked from inside without the need of a key, they don't check the door frame or the lock (which would have revealed that it was bashed open while unlocked) and they do not even consider for a moment the idea that someone could have made a copy of the keys, which is the first thing anyone would have assumed. Not just that; nobody on the floor who has been working at the bank for years says anything about some cleaning lady they haven't ever seen before and that happens to be the first respondant to the murder.
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.