The tune as written here is a weird version of the tune recorded by Dinny Delaney, and seems fairly unique to John Carty. The more standard version (and that recorded by Delaney) is here: https://thesession.org/tunes/11446
Harry Bradley does indeed play the tune on "The Night-Rambler’s Companion", but again plays the more standard version, coupled with Bean Paidín
Re: Dinny Delaney’s
I’m sure that many get this one confused with Slip Jig.