As magistrates, they had their pet points (5)

I believe the answer is:

doges

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'as magistrates they had their' is the definition.
The definition suggests an adverb but the answer is not.

'pet points' is the wordplay.
'pet' becomes 'dog'.
'points' becomes 'es' (E is an example).
'dog'+'es'='DOGES'

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(Other definitions for doges that I've seen before include "Old Venetian magistrates" , "Chief magistrates of Venice and Genoa, formerly" , "Former rulers of Venice" , "Former lords of Venice" .)

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