clutch pedal free play decreases
First you may have gotten oil or grease on the disc or clutch friction plates. Next check your adjustments, the current pedal adjustments may not allow the clutch to sit properly on the disc, as a clutch wears ofetn people will adjust the pedal travel to allow a few more months use out of it, you will nee to back it off.
hydraulic single disc clutch
If the disc is installed correctly the clutch will engage and disengage smoothly. In most cases if the disc is installed backwards the clutch disc will bind against the flywheel/crankshaft bolts and not release resulting with a clutch that doesn't disengage. More than likely if the disc was installed backwards the disc is now damaged and will need to be replaced.
You replace it. The oil is in the clutch face and if you use it, the face will come off of the disc. There is no way to clean it.
no adjustment on hydraulic clutch, worn clutch disc and pressure plate
Clutch disc is worn out. Not knowing what vehicle you have, most are hydraulically operated and there is no adjustment. Time for clutch replacement.
Between the two clutch facings
Clutch disc, pressure plate, transmission shaft,
If it has a hydraulic clutch, I believe they can not be adjusted. When the pedal travel is no longer acceptable the clutch disc must be replaced.
There are various reasons why an oil leak can penetrate a clutch disc. There could possibly be a rear main seal failure. Another possibility could be a failure of the main input shaft seal which could leak towards the clutch disc.
Gear? That is what a clutch does. The clutch assembly which consists of the clutch disc and the pressure plate that applies the pressure to engage the clutch. The clutch on the front of an air conditioner pump is magnetic and engages metal to metal without a clutch disc between them.