In public address, the dear old chap repeats himself (4)
I believe the answer is:
papa
'in public address the dear old chap' is the definition.
I can't tell whether this defines the answer.
'repeats himself' is the wordplay.
'repeats' indicates the doubling of some letters.
'himself' becomes 'pa' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'pa' duplicated is 'PAPA'.
Can you help me to learn more?
(Other definitions for papa that I've seen before include "Father; letter P" , "The old man" , "Dad" , "Daddy]" , "one after Oscar" .)