The building you can see in the picture used to be a pub. Unfortunately, it fell on hard times, was sold, and the building has been converted into student flats.
The name is one I know well. The building is in an area of Plymouth which bears the same name and is quite close to the main railway line from the west country to London. I have passed it many, many times on the train. Apparently, the locals believe that the area was named after the pub rather than vice versa. Wikipedia, however, suggests a “Brythonic derivation of the name Pen y cwm coet, meaning the head of a wooded valley, or Pen y cwm gwyk, referring to the nearby creek, is possible, but must be treated with caution as it is unproven“.
As Sidney Wood points out in a comment on my post about Bessy Beneath Pennycomequick is also a putative earlier name for the town of Falmouth. The Wikipedia article on the town is less cautious about relating this to a Celtic source – Peny-cwm-cuic.
The trouble with all of these derivations is two-fold. First, y as the definite article is Welsh. Cornish has an. Secondly, the word cwm is Welsh. I don’t think Cornish orthography ever used w to represent a vowel sound. Oliver Padel gives comm as the Cornish place-name element meaning “valley” and comments that it occurs very rarely in Cornish names. He hypothesises that the many occurrences of comb or combe in West Country names, either as simplexes or in compounds such as Ilfracombe (Devon) and Wiveliscombe (Somerset) are of English origin. English borrowed the word from Welsh.
So if Pennycomequick does derive from a Celtic source, it would have to be from a name beginning something like pen an comm. The problem is that the Plymouth name is not in Cornwall, but in Devon, and the Falmouth name is only a putative former name, so Pennycomequick does not appear in any of my Cornish place-name reference books.
Photo credit: Google Street View.