"A writing desk" does have something of a Cyberiad-like ambiguity in it!

William

I am unsure if it is a participle or a gerund in this context

I wasn’t sure about this until relatively recently, but it’s actually quite straightforward! It’s a gerund when it functions as a noun, otherwise it’s a participle (i.e. a verbal adjective).

William clearly enjoys writing. [gerund]
I wonder whether he uses a writing desk. [adjective]

I do not know if there should be a hyphen used if written

I only know of it with a hyphen, as in the nursery rhyme that tells of a frog that went “a-wooing”.

I have heard it used, in such as (though I do not know if there should be a hyphen used if written)

"Where's Jim tonight?"

"Oh, he's gone a courting."

And I have heard,

"I'm going a shopping."

When I read or saw about it on television I think there was something about that it was prevalent in the New England part of the United States of America.

Maybe it is something from olden days that has been dropped in some places but not others.

I have tried to find about it on the web but without success yet, because I don't know what the feature is called.

Does anyone know about this please?

William