The engine's output shaft, or PTO (power take-off), has a keyway, and the clutch has a matching one. There should be a tight-fitting key in that keyway, so that the engine can drive the clutch. There are two grub-screws in the clutch body, numbers 2 & 3 in your picture. They are at right-angles to each other. One of them bears on the top of the key, to keep it from moving axially along the shaft, and the other grub-screw bears on the shaft, to keep the whole clutch from moving axially along it. The symptom you describe (shaft will rotate a few degrees in the clutch, but this stops temporarily if you tighten the grub screws) means that either the key is missing, or the shaft and/or clutch is damaged so that the key is now loose-fitting.
You need to remove the engine from the mower, and inspect the PTO, the key, and the keyway in the clutch. If you are lucky, the previous tenant lost the key and has been trying to drive the clutch with the grub-screw by screwing it down into the keyway. If that is all that is wrong you can fix it by cleaning up the mess with a file, then fitting a key. If the correct-sized key is there but has come loose through wear, the problem is more serious. Please post some pictures of the parts after you remove the engine, so we can see whether there is no key, an undersized key, or a damaged key and shaft. Once we know that, we can talk about putting it right.
