Cleaners are a great idea. They serve as a simple introductory enemy and help further the cold war theme of the game by representing the fears of infiltration and subterfuge associated with cold war hysteria. As is they hardly accomplish these goals, being only found in a rare few scripted missions and not being particularly different from any other enemy. I have a few ideas on how to fulfill these goals.
IDEA ONE: "Cleaning" UFO Crash Sites
Why are the cleaners called the cleaners? It seems like their purpose is to hide the existence of the aliens from the world, so a great way to do this would be by having them interfere with UFO crash sites. For the first day after crashing, UFO crash sites remain the same as now. On the second day however the cleaners have arrived. The local forces and civilians spawn as corpses, and there are several cleaner agents spawned around the level according to difficulty making the mission more difficult. On the third day the cleaning is almost complete. The aliens have been evacuated to safety by the cleaners, with all alien enemies replaced by a large number of cleaner soldiers and all alien corpses and technology removed. The disappearance of the UFO after three days can be explained by the cleaners dismantling the craft and removing evidence of the crash.
This change would add urgency to responding to crash sites perhaps motivating players to not always wait out night missions, as waiting a whole day would add agents to make the mission harder and waiting two would remove most of the rewards besides alloys and alenium from the UFO itself, meaning there will be no tech to research or aliens to capture or kill for autopsies and interrogations. This change would also give incentive to eradicate the cleaners, as doing so would permanently end cleaner interference at crash sites. Finally as cleaner soldiers are likely easier to deal with than aliens, this could allow players who have lost all their valuable equipment/good soldiers to possible be able to get back on their feet by taking down mostly cleaned crash sites, which would give them raw materials to build more equipment at the cost of gaining no technology progression.
IDEA TWO: False Flag Attacks
Cleaners could also make combating the aliens more difficult by serving as a distraction and pitting humans against each other. Cleaners will randomly create false flag attacks, especially in regions with low panic. These would serve as mini terror missions taking place in cities, military bases, arctic outposts, etc where you must accomplish tasks like disarming bombs or eliminating all cleaners, or some other unique objective, in a limited amount of turns to prevent a large panic increase and/or regional funding decrease due to the cleaners framing a human nation or faction for the attack and creating a diplomatic incident, distracting the region from their commitment to the Xenonauts.
These missions would create varied conflict outside of scripted story missions like the current cleaner missions. They would also fulfill the niche of missions that can occur even if you are shooting down all the alien UFOs as your air superiority won't have any effect on cleaner activity.
IDEA THREE: Search Teams and Base Defense
The tutorial sets the precedent that cleaners will attack your bases and that you need to stay hidden from them. An interesting way to integrate this into gameplay would be through the cleaners setting up search teams. Randomly cleaner search team headquarters missions will appear in regions with Xenonaut bases, where the cleaners have set up a headquarters for a field team that will counter Xenonaut activity in a region. If the mission is not responded to within a comfortable amount of time (10 days to a month, nothing that urgent), the location of an unrevealed base in your region will be revealed to the cleaners. The Search Team missions themselves are simple, where you have to storm a small cleaner compound in a city, the wilderness, a harbor, etc and kill a VIP. Doing so successfully will end the mission and your base will not be discovered. The usage of the VIP elimination mission type also makes that objective able to appear repeatedly throughout the game instead of just in a few scripted missions, building variety.
Cleaner base attacks will function differently from alien ones. While aliens attack as retribution for shooting down UFOs, cleaners will attack any base they know about without provocation. This means bases without hangars are not safe which gives you an extra incentive to respond to search team missions that might reveal them, and also to build defenses and station soldiers at every base you have.
IDEA FOUR: More Cleaner Enemy Types
Cleaners are supposed to just be regular humans, and an interesting way to use this concept would be to have them mirror the Xenonauts in their combat style. Cleaner enemies could come specialized into the general roles available to the Xenonauts. While cleaners would stay as one enemy type they could be equipped with a wider range of weapons. Cleaner riflemen would make up the majority of cleaner forces and be the same as current cleaner soldiers. Cleaner snipers would stay still and keep a distance from enemies. Cleaner shotgunners could throw smoke grenades and shoot your soldiers at close range. Cleaner heavies could wear heavy armor and use a machine gun to suppress your troops with low accuracy fire. Cleaner grenadiers could destroy cover used by your soldiers to force you out of position. Cleaners with shields would throw lethal grenades and need to be flanked similar to a cyberdrone, but without automatically turning on hearing sound. Finally a MARS like vehicle operated by the cleaners could serve as a boss type enemy with a cannon or rocket launcher and autocannon. Finally to emphasize their mind controlled nature cleaners could be immune to morale failures when an alien serving as a commander is nearby (As a reference to the VIP cleaner cell missions where aliens are in the field giving orders to the cleaners), and most major cleaner missions would have a Psyon present serving as this commander, with the weakness of having very low morale when a commander isn't alive, giving incentive to pick the commander off early, and making cleaners in minor missions without commanders like UFO crash sites easier to deal with.
These changes would help to make cleaners a more varied enemy to fight and a proper faction in their own right distinct from the aliens. The cleaners with special roles would be rare, making up only a small portion of cleaners in missions who would still mostly be the default rifle goons for simplicity, and would be introduced gradually with the cleaners gaining their whole arsenal of troops around the beginning of the midgame, as cleaners are supposed to serve primarily as early game enemies.
IDEA FIVE: Cleaner Data
While slowly revealing the cleaner cells and then the cleaner base that you can take down to end the cleaner threat for good is a solid idea, progression towards ending the cleaners is automatic and doesn't feel very rewarding, more like waiting out a timer than making progress. A solution to this is to make the cleaner progress only increase upon the acquisition of cleaner data, and to have cleaner data present in all cleaner missions. The starting office mission could remain the same, with getting cleaner data as a mandatory objective and to introduce players to the concept, but cleaner data could be present in all cleaner missions. Search Team Headquarters mission maps will have optional cleaner data items accessible from computers. Alternatively, cleaner progress can be increased directly if you capture cleaners alive and then interrogate them for a small progress bonus or capture and interrogate a Psyon cleaner commander for a much larger amount of progress.
The final mission would function similarly, becoming available after reaching 100% cleaner progress, but an interesting idea would be to make it a two part mission like the final mission of the first X-COM game on Cydonia, with one mission on the arctic surface and the second portion in the underground facility, as the added difficulty would match the increased significance of ending the cleaners once and for all due to their increased importance.