Son beginning to correct his and my pronouns -- it's a project (6)
I believe the answer is:
scheme
'it's a project' is the definition.
(I've seen this in another clue)
'son beginning to correct his and my pronouns' is the wordplay.
'son' becomes 's' (genealogical abbreviation for son).
'beginning to' indicates taking the first letters.
'his and my pronouns' becomes 'he me' (object pronouns corresponding to 'him' and 'my').
The first letter of 'correct' is 'c'.
's'+'c'+'heme'='SCHEME'
(Other definitions for scheme that I've seen before include "Connive, intrigue" , "'Plot, plan (6)'" , "Wheel and deal" , "Plan - intrigue" , "Conspire" .)