Revealing mistake: Watch the opening sequence of this episode very carefully - Clarkson, May and Hammond are not in Downing Street (they are on a set that stands on the BBC backlot) and library footage of the Prime Minister is cleverly edited in to make it look as if they were there at the same time. Even taking into account the fact that the Prime Minister's security service would never allow such an intrusion onto Downing Street, the line up of the shots does not quite match - our three intrepid heroes should be visible in shot when the Prime Minister is shown exiting number 10, but they are nowhere to be seen. It is all sleight of hand and clever editing, but detectable if you watch carefully enough.
Patagonia Special: Part 1 - S21-E8
Revealing mistake: On their boat trip to Tierra del Fuego there are a number of sweeping overhead shots of the ferry upon which Clarkson, Hammond and May are travelling. Their very distinctive cars are not on deck. There is nowhere on the only deck that they could be under cover. The footage of the ferry was filmed on a different voyage.
Revealing mistake: Jeremy Clarkson remarks early on in the episode (during the journey to Da Lat) that he would rather "hop to Hanoi." Later in the episode, the trio make a mistake whilst purchasing their train tickets and accidentally travel to Hanoi rather than Ha Long City. The comment made suggests that the plan was to travel to Hanoi all along and the "mistake" was scripted.
Suggested correction: This isn't something they hide. Behind the scenes interviews make it quite clear that they have a B unit that films additional footage of the areas they travel in. They even take the cars back to get more shots of them on the road, since they can't predict the exact routes that will be taken in order to have camera crews ahead of them in time.
The mistake is perfectly valid. The correction is only providing an excuse as to why the mistake occurs. Most shows/films have a second unit director and crew (the "B unit") solely to film supplemental footage. This does not invalidate the point of the mistake, that "their very distinctive cars are not on deck" where they should be.
Super Grover ★
Since they cut to and from shots of Hammond, May and Clarkson standing on the deck of the boat the implication is clear - the shots are continuous. It is a glaring error.