Involved old Communist having change of heart (3,2)
I believe the answer is:
let in
'involved old communist' is the definition.
'let in' can be an answer for 'involved' (to let someone in to a plan is to involve them). I'm not certain of the 'old communist' bit.
'change of heart' is the wordplay.
'change' becomes 'let' (let is a kind of change).
'of heart' becomes 'in' (I am not sure about this - if you are sure you should believe this answer much more).
'let'+'in'='LET IN'
'having' is the link.
Can you help me to learn more?
(Other definitions for let in that I've seen before include "Grant entry" , "Admit" , "Allowed to enter" , "Allow to enter" , "swindle . . ." .)