The mark on the top of the key has been made by the grub-screw. This is normal and appropriate: the screw has to bite into the key to securely keep it from sliding longitudinally in service. However the key should be a firm fit, in both keyways - the external one in the shaft, and the internal one in the clutch bore. If it is a slack fit laterally, it will not work properly.
The engine appears to be a Briggs and Stratton, which is built to inch dimensions, not metric. A 3/16" key would be 4.76 mm square, not 4.4 mm. I expect this would apply equally to the Ransome mower, which I believe would have been made in Britain some years ago, presumably using inch dimensions. If you actually have a 4.4 mm square key, it is probably not the original one, and not of the required size.
