Plum Prime Minister dropped by police branch - that's understandable (5)

I believe the answer is:

lucid

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'that's understandable' is the definition.
(synonyms)

'plum prime minister dropped by police' is the wordplay.
'prime minister' becomes 'PM' (abbreviation).
'dropped by' is a deletion indicator.
'police' becomes 'cid' (abbreviation for Criminal Investigation Department).
'plum' with 'pm' removed is 'lu'.
'lu'+'cid'='LUCID'

'branch' is the link.
I am not very happy about this link. It may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for lucid that I've seen before include "Shining, clearly" , "Limpid" , "Thinking clearly" , "Coherent" , "Transparent" .)

I've seen this clue in the Sydney Morning Herald.

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