I'm no Dubliner, but I can try and help you out. Hopefully you'll get an answer from someone that is Irish and knows what they're talking about. I think twang means toffee, as far as I remember. Hawkin' is an abbreviation of Hawking and means selling (to sell). He was selling toffee. I think that the Traycle depot is some kind of shop that she owned near Carlisle Bride, but I'm not 100% sure about what kind of shop. Maybe a Treacle shop, or a sweet shop? That is a guess though. Just so you know this is a hard song to translate because it's full of a lot of very Dublin specific slang.

THE TWANGMAN (C.O'L.,1, p.231) — A lighthearted murder story from the middle of the last century. I am told that twang man was a kind of sweetmeat (Partridge's definition cannot be admitted to holy Ireland) and that Treacle Billy was a kind of toffee made from treacle. Carlisle Bridge is now O'Connell Bridge.