In my opinion, it is a mystery why the Klingon Empire can't simply relocate the population of the Home planet, which should be merely one out of several thousand, million, or billion inhabited and civilized planets in the Empire, to other inhabited and civilized planets in the Empire.
In "Errand of Mercy" Kirk tells the Organians:
KIRK: Gentlemen, I have seen what the Klingons do to planets like yours. They are organised into vast slave labour camps. No freedoms whatsoever. Your goods will be confiscated. Hostages taken and killed, your leaders confined. You'd be far better off on a penal planet. Infinitely better off.
So Kirk apparently has seen planets with native populations ruled by the Klingons, and probably producing products for Klingons to use.
And Kor expects that the klingons are about to conquer many more planets, which will no doubt include many planets where they can live as comfortably as on Organia:
KOR: Sentimentality, mercy. The emotions of peace. Your weakness, Captain Kirk. The Klingon Empire shall win. Think of it, as we sit here, in space above us the destiny of the galaxy will be decided for the next ten thousand years. Can I offer you a drink? We can toast the victory of the Klingon fleet.
And Kirk tells the Organians:
KIRK: We have legitimate grievances against the Klingons. They've invaded our territory, killed our citizens. They're openly aggressive. They've boasted that they'll take over half the galaxy.
There are hundreds of billions of star systems in our galaxy. So if Kor and Kirk exaggerated the number of stars that the Klingons hoped and expected to conquer from the Federation by only a thousand times, the Klingons would still be expecting to conquer hundreds of millions of star systems with their planets. And the Klingons would need to already rule hundreds of thousands or millions of inhabited planets to have any expectation of conquering hundreds of millions of planets.
Suppose that Kor and Kirk exaggerated the number of stars and planets that the Klingons expected to conquer by a million times. Then the Klingons would be expecting to conquer "only" a "mere" few hundred thousand stars. And the Klingons would need to already rule hundreds or thousands of inhabited planets to have any expectation of conquering hundreds of thousands of planets.
In "A Private Little War" a Klingon named Krell, who seems to be comfortable enough on a class M planet, tells his local ally Apella:
KRELL: You will be rich one day, Apella, beyond your dreams. The leader of a whole world. A governor in the Klingon Empire.
In "The Trouble with Tribbles", "Friday's Child", and "A Private Little War" Klingons seem to have no trouble breathing the air and otherwise living on class M planets suitable for humans, which seem to be very common in Star Trek. So if the Klingons rule a large region of space they are likely to rule many planets suitable for them to live on, as well as ruling many subject planets which produce products for Klingon use.
In "Day of the Dove" Mara says:
MARA: We have always fought. We must. We are hunters, Captain, tracking and taking what we need. There are poor planets in the Klingon systems, we must push outward if we are to survive.
This implies that Klingons live on several worlds, and that they need imports to survive, even before Praxis explodes, and that their main export that they pay for imports with is life; i.e. they don't kill people who provide those imports.
The Klingons still rule an unspecified number of planets about a century later in the TNG episode "The Mind's Eye":
Captain's log, stardate 44891.6. The Enterprise has been ordered to accompany a special emissary from the Klingon High Council to the Kriosian System, where one of their colonies is fighting for independence.
KELL: There was a time when the Empire would crush a rebellion. Now it is tolerated. We have enough problems on the home planet. We don't wish to divert resources to such a trivial war.
The rebellious people on Krios might be a non Klingon species, or Klingon colonists, or a mixture of both.
In Star trek V: The Final Frontier the history of Nimbus III, the Planet of Galactic Peace is described briefly.
[Paradise City saloon backroom]
CAITHLIN: Gentlemen, I'm Caithlin Dar.
TALBOT: Ah, yes. Our new Romulan representative. Welcome to Paradise City, my dear, capital of the so-called 'Planet of Galactic Peace.' I'm St. John Talbot, the Federation representative here on Nimbus Three and my charming companion, here, is the Klingon consul Korrd.>
CAITHLIN: I expect that's Klingon for hello.
TALBOT (OC): Won't you come in, my dear?
(Sybok and his followers approach Paradise City)
AITHLIN: Twenty years ago, our three governments agreed to develop this planet together. A new age was born.
TALBOT: Our new age died a quick death. And the settlers we conned into coming here, they were the dregs of the galaxy. They immediately took to fighting amongst themselves. We forbad them weapons, but they soon began to fashion their own.
CAITHLIN: Right! Then it appears I've arrived just in time.
The population of Nimbus III includes many species which were not seen previously, who could be from various Federation, Romulan, and Klingon worlds.
The Klingon prison planet Rura Penthe in Star trek VI: The Undiscovered Country has Klingon guards but the prisoners all seem to be members of non Klingon species. I sort of doubt that they were all foreign travelers who committed crimes in the Klingon Empire, so most of them should be members of subject species from subject worlds in the Klingon Empire.
It seems to me that if the Klingons move to many different planets in their empire and thus move closer to the producers of products that used to be sent to their home world, they will eliminate the shipping costs for those products and thus could reduce their taxes on their subjects while increasing the goods they receive from the subjects. If the klingons have a population of ten billion on their home world, and send ten million Klingons to each of a thousand technologically advanced subject planets which have average populations of one billion, that will result in a mere one percent population increase on each subject planet, which should be something they can handle fairly easily.
So supporting a relocated Klingon home world population should be easy for the Klingon empire unless they rule only a small number of planets. And if the Klingon Empire has to shirk in size to rule only a small number of planets, then the other Alpha and Beta Quadrant powers that border the Klingon empire, like the Federation and the Romulan Empire, also have to shrink in size to explain why the Klingons haven't been totally defeated, conquered, and annexed already.
So the various Star Trek productions give very contradictory indications of the size of various space governments that are depicted.
See also my posts number 8 and 21 in this discussion, https://www.trekbbs.com/threads/who-should-the-next-rival-of-the-federation.303030/1 for more information about the puzzle of the sizes of the various powers in Star Trek.