As it stands the Cleaners are a neat idea similar to EXALT and ADVENT from the nuXCOM games, but they're fairly underutilized. Their presence is limited to 4 missions (5 if you count the tutorial): the mission where you steal 10 data sticks, the 2 regional command VIP assassinations, and then the final assault on the Cleaner base. They've also only got the 2 enemy types: the suited agents and the hazmat-wearing soldiers.
So here are some suggestions on expanding not only their presence in the early game but also mission variety:
VIP Rescue Mission
Your Operations Director identifies a heavily guarded compound where a certain VIP is being held by the Cleaners. Her agents are unable to infiltrate or extract the VIP, thus it falls upon you and your heavily armed soldiers to perform a prison break. As the VIP is being held at the facility for only a certain window of time, the mission has a time limit on geoscape. The compound would have a mix of Cleaner soldiers and agents as guards, with reinforcements incoming as with the regional command missions. The VIP would be held in a similar cell thing as the civilians in alien abduction missions, and once you free them they can be ordered around like your soldiers. Optional objectives could be gathering more data sticks and perhaps rescuing other identified Persons of Interest. Victory is when you extract your team along with the VIP and any PoI. Rewards would be panic reduction in the region the mission is in, increased with each PoI rescued.
Sting Operation/Convoy Interception Mission
Your Operations Director's agents either located the route a Cleaner convoy carrying valuable alien equipment is traveling through or set up a trap for a Cleaner strike team. Either way, there's a limited amount of time to intercept them. The enemies would be a large amount of Cleaner soldiers, with the convoy perhaps having a few lower ranking aliens like Sectons or Sebillians hidden away. Like base defense, you would be able to set up the position of your soldiers within certain areas and had the entire map revealed as you have already scouted out the area and set up the ambush, however you would still have the first turn. Victory is from killing or incapacitating all enemies, with the rewards being either some alien alloys and alenium from a convoy interception or a regional panic reduction and Cleaners network progress for sting operation.
Cleaner presence on crash sites
Right now the Cleaners not only don't really live up to their name as "Cleaners" but a crash site with 1 hour left on the timer is no different from one with 2 days left. It would make sense that the Cleaners clean up the crash sites and make it so delaying a crash site has consequences. Perhaps for each day that passes, there is more Cleaner presence at the crash site, with more of the ambient civilians replaced by 1 or 2 Cleaner soldiers. This would go away once the Cleaners' base is destroyed, but adds some more urgency in the early game to take on crash sites.
Cleaner Heavy
Simple, a Cleaner soldier that uses a machine gun (same stats as ours) and is heavily armored. The look can borrow from EOD uniforms or other heavily armed military gear such as this, this, or this. They'd only appear on missions with a heavy Cleaner presence such as sting operation or convoy interception or the Cleaner base defense, or perhaps as part of later reinforcement waves to really force the player to extract.
Would like to hear others' thoughts and suggestions along these lines!