If you mean https://www.openstreetmap.org/way/1442691542, it is already correctly tagged as access=private, and hence displays in a faint grey on the map.

If you mean https://www.openstreetmap.org/way/491169745 then that is a public footpath (according to the definitive map) and is tagged as such.

So I think the mapping is correct here. Please reopen this note if I’ve misinterpreted, thanks.