Is this really some kind of 'road' maintained by the council?
On OS Streetview and even old OS maps, they show it as perhaps some kind of track.

No longer called Little Henry's.
Seems to be now a upmarket bistro/restaurant rather than a cafe.
I don't remember the name

Is this footpath closed / actually still there due to coastal erosion?
There's a closed sign at eastern end, but that might apply just to the little bit at the corner.
Access no has been added to a short bit on the western end.
NB Hampshire CC website does not mention this path as closed.

Anyway you can just walk along the beach instead, even at high tide.