The old man's at half pace again (4)
I believe the answer is:
papa
'the old man's' is the definition.
('papa' can be a synonym of 'old man')
'at half pace again' is the wordplay.
'at' says to put letters next to each other.
'half' suggests halving the letters.
'again' becomes 'pa' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'pace' cut in half is 'pa'.
'pa'+'pa' is 'PAPA'.
Can you help me to learn more?
(Other definitions for papa that I've seen before include "Mama's other half?" , "Comms code for 'P'" , "Pop" , "The old man" , "Father; P (radio comms.)" .)