Domestic doctor has a student (5)
I believe the answer is:
valet
'domestic' is the definition.
(a domestic servant)
'doctor has a student' is the wordplay.
'doctor' becomes 'vet' (vetting is a kind of doctoring).
'has' indicates putting letters inside.
'student' becomes 'l' (as in L-plates for learner drivers).
'a'+'l'='al'
'vet' going around 'al' is 'VALET'.
(Other definitions for valet that I've seen before include "Gentleman's assistant" , "Personal servant" , "Man's personal attendant" , "Gentleman's gentleman" , "Gentleman%u2019s gentleman" .)