David Johnson has kindly replied to my query, as follows:
Let's start by discounting various suggestions made:
1. lime kiln - absolutely not for various reasons
2. connected with peat cutting - not up there and I see no connection between the feature and anything to do with turbary
3. storage - it's just too high up, too damp and far too remote for this
4. shelter - too small in all dimensions. There are several such shelters below The Arks and one below Gragareth and umpteen elsewhere and they are all very different from this feature
5. rabbit smoot - a rabbit smoot is a low passage through the base or near the base of a drystone wall most often originally with a trap on one side to catch bunny unawares
I hesitate to say what I think it might be without having seen it in the flesh and, more importantly, seen it in its wider context so if someone can let me have an 8 or 10-figure grid reference I will go up sometime and have a look for myself. However ... and this remains a hypothesis at the moment ... I have surveyed on the ground (and published) six long leats or cut channels that fed water in a controlled manner from the fells to particular points of use - farmsteads or water mills. The engineering employed to create them is awe inspiring - to get the water flowing but not too fast. One fed Burton in Lonsdale water mill and Bideber water mill en route; one fed Philpin off Ingleborough; one fed Bleak Bank off Little Ingleborough; one fed Know Gap and Clapdale off Little Ingleborough; one fed the Scales farms at C-le-D off the base of Whernside near Ellerbeck; and one fed Broadrake off Whernside. The feature under discussion is too far 'off course' to have fed Broadrake's leat, and I have found no trace of it on the hillside, but I wonder if the feature was the start of a leat feeding Scar Top (formerly Top Farm) in the past. To ensure they worked well throughout the year the source point(s) were carefully designed and maintained.
David has a lot on his plate at the moment but he said he'll try and get up there at some stage and let me know what he thinks, so I can post further information on here. (I've explained exactly how to find it.)
He mentioned Scar Top; that was where the late Albert Chapman (YRC) lived for several decades. I knew Albert well and I'm sure he'd have been able to shed some light on this mystery walled entrance. He used to be up on Whernside all the time, often using it to train for Himalayan adventures. Sadly, we lost Albert not so long ago.
He also mentions artificial water courses elsewhere and in the vicinity. Regular forum users may remember we had a discussion topic about Grey Wife Sike and Know Gap Sike (on Ingleborough) during Covid lock down:
Here's one to exercise your grey matter (no pun intended) whilst we're all grounded during the coronavivus lockdown. Grey wife Sike (a small stream west of Cote Gill) was artificially channelled in the past, to avoid it sinking in various shakeholes where it meets the Great Scar Limestone at...
ukcaving.com
Anyway, thanks David; we'll look forward to learning more, when you get the chance to take a look.