What could be sigh or dry, rough cough? (3,5)
I believe the answer is:
eye rhyme
'rough cough?' is the definition.
('rough' and 'cough' are eye rhymes)
I can't explain the remainder of the clue.
(Other definitions for eye rhyme that I've seen before include "Similarity between words in spelling but not in sound" , "Similarity in spelling but not sound as in love and move" , "Maid said, for example" , "shy?" , "scan piece of poetry" .)