Without source, fop twice claiming party's leader is BBC puppet (4,5)
I believe the answer is:
andy pandy
'bbc puppet' is the definition.
I know nothing about this answer so I can't tell whether it can be defined by this definition.
'without source fop twice claiming party's leader' is the wordplay.
'without source' means to remove the first letter.
'fop' becomes 'dandy' (synonyms).
'twice' shows that letters should be duplicated.
'claiming' indicates putting letters inside.
'party's leader' becomes 'p' (I am not sure about this - if you are sure you should believe this answer much more).
'dandy' with its first letter taken away is 'andy'.
'andy' duplicated is 'andyandy'.
'andyandy' going around 'p' is 'ANDY-PANDY'.
'is' acts as a link.
Can you help me to learn more?
(Another definition for andy pandy that I've seen is " TV character".)