Seem truthful, but have police record? (5,10)
I believe the answer is:
carry conviction
'seem truthful' is the definition.
(to carry conviction is to be convincing)
'have police record?' is the wordplay.
'have' becomes 'carry' (carrying is a kind of having).
'police record?' becomes 'conviction' (conviction for a crime).
'carry'+'conviction'='CARRY CONVICTION'
'but' is the link.
(Other definitions for carry conviction that I've seen before include "to be believed" , "credibility" , "Criminal may have" .)