Slowly introduce prayer for dead fellow involved in action (4,4)
I believe the answer is:
drip feed
'slowly introduce' is the definition.
(to drip feed something is to introduce it slowly)
'prayer for dead fellow involved in action' is the wordplay.
'prayer for dead' becomes 'RIP' (rest in peace).
'fellow' becomes 'F' (abbreviation - of a society etc.).
'involved in' is an insertion indicator.
'action' becomes 'deed' (deed is a kind of action).
'rip'+'f'='ripf'
'ripf' inserted into 'deed' is 'DRIP-FEED'.
(Other definitions for drip feed that I've seen before include "Drop supplies" , "supply in small instalments" .)