Angela, the only soldier I've seen with a scruffier uniform than your grandfather's is my grandfather's! <s> He could never have got away with it in the Military Police Corps. Plus I don't recall ever seeing a redcap (so-called because when on duty a red cover is worn on the cap) who had less than two stripes! The single stripe indicates the rank of lance-corporal or lance-bombadier in units other than the MPs.
But the white webbing (belt, gaiters, etc) does suggest a police function within his unit, whatever that was. The cap badge would reveal his corps or regiment, if only it could be enlarged sufficiently.
Wikipedia has a picture of the current badge of the Royal Military Police, which is different from your grandfather's. Good luck in your search.