The British-English phrase to drop a clanger means to make an absurd or embarrassing blunder—cf. also to drop a brick.

The noun clanger is used only in this phrase; the image is of something dropping with a clang, i.e. with a loud resonant ringing sound, which underlines the conspicuous nature of the mistake.

This image is clear in an early instance of the phrase, used as the punning title of an article published in the Sevenoaks Chronicle, Westerham Courier and Kentish Advertiser (Sevenoaks, Kent) of Friday 7^th April 1950:

DROPPED A CLANGER!

On Monday, workmen in Linden Square, Riverhead, were startled to hear a very loud clanging from the main roadway. So loud was it that they rushed to see what had happened.
They found that a heavy lorry had shed the whole of its propellor shaft in the road. The driver had to wait helpless, with nothing wrong with his engine, until a mate came to tow him away.

This phrase seems to have originated in British Army slang during or immediately after the Second World War; the earliest instance that I have found is from Beeze Up Your English!, published in the Daily Herald (London) of Monday 10^th September 1945:

First soldier (surveying his dinner plate): “Slingers and Gippo again!”
Second soldier: “What’s for duff?”
First soldier: “Burma Road! What a shower!”
It was overheard the other day among men with the British Forces in Germany.
That’s the way they talk in the Army now. So here’s your dictionary to interpret it when the boys come:
Slingers and Gippo—Sausages and gravy.
Burma Road—Rice pudding.
What a shower—Not going to like it.

Look Jilty¹

It is so easy for the innocent civilian to drop a clanger (make a mistake) that this further page from the Army dictionary should be useful:
On stag—On guard.
A snout—A cigarette.
Ling him one up—Salute him.
Get a rocket back—Ticked off.
Up for office—To see C.O.²
Beezing—Polishing.
On a fizzer—Under open arrest.
Look jilty¹—Get a move on.
Charp it out—Get to sleep.
So be on stag when he comes home and asks for a snout. It’s Army English!

¹ The noun jilty (also jeldi, jildy, and other variants) means haste—origin: Hindi jaldi, quickness.
² C.O.: Commanding Officer

Later in 1945, several Canadian and U.S. newspapers published an article distributed by The Canadian Press (CP), apparently inspired by the British article that I have reproduced, and containing a few transcription errors; for example, on Saturday 17^th November 1945, The Ottawa Journal (Ottawa, Ontario) published Army Slang Baffles Civilians:

London, Nov. 17.—(CP)—A brand-new threat from Germany is reported to be causing some alarm in British households. British soldiers, already notorious for their slang, are importing some post-war doubletalk that seems designed specifically to baffle civilians.
A sample conversation in this latter-day Army English might go something like this:
First soldier (surveying his dinner plate): “Slingers and Gippo again!”
Second soldier: “A bit ropey. What’s for duff?”
First soldier: “Burma Road! What a shower!”
Glossary for the family:
Slingers and Gippo—sausages and gravy.
A bit ropey—poor cooking.
Burma Road—rice pudding.
What a shower—expression of disgust.
Ma and Pa, if they don’t want to drop a clanger, will also have to beeze up on their English with the following glossary:
Ling him one up—salute him.
Look jitty [sic]—get a move on.
Charp it out—get to sleep.
On a fizzer—open arrest.
Up for office—see commanding officer.
Beezing—polishing.
Drop a clanger—make a mistake.
It is also wise for the family to be on slag [sic] for Tommy might ask for a grout [sic]; i.e. on guard, for he might try to borrow a cigarette.