When Dot threatens Hugh with the gun, the pencil he holds in his right hand disappears and reappears several times between camera angles.
At the end of the previous episode "Murder in the Dark," Foyle cuts the phone line at Phyrne's home. At the beginning of this episode, which apparently takes place the next day, Burt goes to make a phone call to the police. Though we don't see his actions past picking up the receiver, the police appear at the scene as directed, suggesting no phone trouble, but surer evidence of a continuity error comes when, with no mention of the phone line ever needing repair, Phryne calls the house: The phone is shown ringing repeatedly with no one answering.
At around 4'20", the victim is seen with his mouth slightly open, but in the next shot his mouth is completely closed.
In the Southern Hemisphere, where the series takes places, Mid-Summer's Eve would be December 21. However, King Memses, from the Northern Hemisphere, would have celebrated Mid-Summer's Eve on June 21, if, at all.
Several mentions of Foyle's archaeological dig in Egypt make it sound as though the Valley of the Kings (where "King Memses' Tomb" was) is right near Giza (where the famous pyramids are). In fact, they are over 600 km apart.