Hinny is a term of endearment, honey (possibly) though it carries a very different weight in ordinary usage - i.e. men will often call friends hinny by never honey!

We sing it thus (i.e without Johnny Handle's additions):

The kye come hyem but aa saw not me hinny
The kye come hyem but aa saw not me bairn
Aad rather lost aa the kye than lost me hinny
Aad rather lost aa the kye than lost me bairn

Fair faced is me hinny his blue eyes a shining
His hair in gold ringlets hangs sweet to me sight
So off ye gan daddy and look for your laddie
And bring tiv his mammy some peace tiv her mind

He's aalways out roamin the lang summers day through
He's aalways out roamin away from the farm
Through hedges and ditches and valleys and fellsides
I hope that me bairnie has come to nee harm

I've searched in the meadow and in the fower acre
Through stackyards and byres but nowt could aa find
So off ye gan daddy and look for your laddie
And bring tiv his mammy some peace tiv her mind

The kye come hyem but aa saw not me hinny
The kye come hyem but aa saw not me bairn
Aad rather lost aa the kye than lost me hinny
Aad rather lost aa the kye than lost me bairn