There were actually 2 children of David and Martha, that received the name Handley. The first was born in Mar 1855 and baptised shortly afterwards. Sadly, this Handley died in August the same year and is buried in Ashford Cemetery, Derbyshire.
Number 2 Handley was born Ashford in the Water 22 Jun 1856. He was only a little baby when he stepped onto the ‘Josepher Fletcher’ heading towards New Zealand.
As far as I have been able to work out, Handley died at sea somewhere in the Southern Ocean below Australia, a matter of a week or so from landing at Lyttelton. His little body would have been given up to the sea.

Memorial to his namesake younger brother and himself with their father’s grave at Avonside Church, Christchurch. It’s interesting to ponder, but perhaps the first Handley did not have a gravestone in England, so was captured here. We are unlikely to ever know.
Difficult to read, but you can see the two Handley boys above Elizabeth (DW Hamilton’s second wife). His first wife, Martha is at the top, but her body is actually in Lyttelton Cemetery. Photo from Elizabeth Weissenstein.