Earnestly request quiet bit of sunshine (4)
I believe the answer is:
pray
'earnestly request' is the definition.
(pray can mean to request)
'quiet bit of sunshine' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'bit of sunshine' becomes 'ray' (a ray of light).
'p'+'ray'='PRAY'
(Other definitions for pray that I've seen before include "Appeal to higher authority" , "Earnestly ask (God?)" , "Address a deity" , "Wish earnestly (for a particular outcome)" , "Supplicate, entreat" .)