Historically, a client state is basically a state ranging anywhere from being a reluctant allies to being a puppet state. The idea comes from the Roman Empire.
The idea is that your kids are taken to be educated / hostages and are educated in the Imperium, they come back and eventually come to rule their nation as good allies who have adopted the Roman mindset. In return for all of this, the Romans will trade with you and will likely protect you from foreign aggressors. How they actually felt about being Clients varied from nation to nation, but most were walking a tightrope - they were basically not conquered by Rome because Rome didn't want to put in the effort. Most client states ended up being annexed by Rome at some point, often within a generation or two of becoming clients.
The Imperium's Clients seem a bit different. There are some large scale maps of Charted Space where you see "Imperial Client States" who are on the other side of the Zhodani Consulate (and lots of them). I have no idea how that Imperium can really offer much of anything to Clients way out there - the Imperium certainly isn't going to go to war with the Zhodani over them. I suspect they're out there because someone decided to interpret "Client State" as "allied nations" or "bloc nations" from the 21st century.
Given their situation, it's very likely an Imperial noble would be given courtesy while in a client state - insults or outrages to an Imperial noble by foreigners is one of those things the Imperium isn't going to look too kindly on.
Similarly, an Imperial-educated ruler of a planet might miss how things were during his or her school days in Capital and might want to surround him/herself with Imperials to sort of relive the old days so might grant an Imperial noble a fief on a world. Of course, there's always the concern that such a foreign educated ruler is out of step with the actual natives of the world, who might feel like their king/queen is outsiders come and govern them. This could cause considerable resentment. On the other hand, the fief might be granted in a place where few if any people live. The ruler might also harbor some desire for independence from the Imperium so might grant a fief as a kind of poisoned chalice - assigning the player a fief that is deliberately difficult to tame or ungovernable in some way in the hopes of showing the Imperium in a bad light to the locals. Or the local people might be enthusiastic about an Imperial noble.
