Low offer to pen opener for rock singer (8)
I believe the answer is:
bluebird
'singer' is the definition.
(bluebird is a kind of birdanimal)
'low offer to pen opener for rock' is the wordplay.
'low' becomes 'blue' (both can mean unhappy).
'offer' becomes 'bid' (bid is a kind of offer).
'to pen' indicates putting letters inside.
'opener for' indicates taking the first letters.
The initial letter of 'rock' is 'r'.
'bid' going around 'r' is 'bird'.
'blue'+'bird'='BLUEBIRD'
(Other definitions for bluebird that I've seen before include "North American songbird (once seen over Dover?)" , "symbol of happiness" , "Cerulean flier associated with happiness" , "Fast car - or boat" , "US flier (but never seen over the white cliffs of Dover)" .)