After short time children's author becomes vulgar (6)
I believe the answer is:
tawdry
'vulgar' is the definition.
(synonyms)
'after short time children's author' is the wordplay.
I cannot quite see how this works, but
'time' could be 't' (abbreviation) and 't' is found in the answer.
'author' could be 'awdry' (I've seen this before) and 'awdry' is present in the answer.
No letters remain.
This explanation may well be incorrect...
'becomes' acts as a link.
Can you help me to learn more?
(Other definitions for tawdry that I've seen before include "Shabbily gaudy" , "Tastelessly showy" , "In poor taste" , "Showy, cheap; sordid" , "Gaudy but valueless (6)" .)