Put up with a troublesome pet catching cold repeatedly (6)
I believe the answer is:
accept
'put up with' is the definition.
(accepting is a kind of putting up with)
'a troublesome pet catching cold repeatedly' is the wordplay.
'troublesome' indicates an anagram.
'catching' means one lot of letters goes inside another.
'cold' becomes 'c' (eg on taps).
'repeatedly' shows that letters should be duplicated.
'pet' anagrammed gives 'ept'.
'c' doubled is 'cc'.
'ept' placed around 'cc' is 'ccept'.
'a'+'ccept'='ACCEPT'
(Other definitions for accept that I've seen before include "Swallow" , "Receive voluntarily" , "Believe to be valid" , "Bear" , "Regard with approval; take on" .)