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.