Anniversary gift using a third battery (8,5)

'anniversary gift' is the definition.
(gift for the anniversary of someone's birth)

'a third battery' is the wordplay.
I cannot quite see how this works, but
'a' is found in the answer.
an anagram of 'third' is 'irthd' which is located in the answer.
an anagram of 'battery' is 'bytreat' which is within the left over letters.
This accounts for all the letters.
This explanation may well be incorrect...

'using' is the link.