Frequently scrubbed droopy tail of Rex (5)
I believe the answer is:
royal
'of rex' is the definition.
('rex' is Latin for king)
'frequently scrubbed droopy tail' is the wordplay.
'frequently scrubbed' means one should take alternating letters (letters removed or scrubbed at regular intervals).
'droopy'+'tail'='droopytail'
The alternating letters of 'droopytail' are 'ROYAL'.
(Other definitions for royal that I've seen before include "Member of monarchy" , "As kings and queens" , "Wills, perhaps" , "Relating to king/queen" , "Relating to a monarch" .)