Continuity mistake: When Jimmy reads the solicitor's letter, the date is 5 October 1935. Hastings wrecks his car here but in Murder in the Mews, taking place on the 6th of November, he has it again. Assuming that he has been able to get his car back to form entirely restoring/replacing the ruined sections (Japp tells him that it's now just "an expensive scrap of metal"), in that episode the car needed ordinary maintenance work, while going by the events of this episode it would have been fresh off the car shop instead.
Character mistake: As Poirot reconstructs the facts in the flat (and he explicitly says "consider the FACTS"), he lays out that "a letter which was found at the scene of the crime with John Frazer written on the bottom." But the letter simply said "Frazer", the first name was just Japp's random conjecture based on the initials on another evidence being J.F. (00:27:40)
Continuity mistake: As Poirot stands from the late dinner with the young socialites to go talk to Inspector Japp next to the door, he leaves his napkin on the table. The napkin changes between shots, going from mostly straight to crumpled (when Mildred says "How dreadful!") to straight again. (00:22:20)
Continuity mistake: As the two porters are carrying the piece of furniture, the draping on top of it goes all the way down one side when they have just unloaded it from the van, and the opposite side when they are bringing it up Whitehaven Mansions' stairs getting cranky at the doorman. (00:01:40)