Crossword Genius

Cough me up a letter! (3)

Ross

I believe the answer is:

hem

I'm a little stuck... Click here to teach me more about this clue!

'a letter' is the definition.
Although both the answer and definition are singular nouns, I can't understand how one could define the other.

'cough me up' is the wordplay.
'cough' becomes 'h' (I can't justify this - if you can you should give a lot more credence to this answer).
'up' is a reversal indicator (in down clue: letters go upwards).
'me' back-to-front is 'em'.
'h'+'em'='HEM'

Can you help me to learn more?

(Other definitions for hem that I've seen before include "Double-backed cloth border" , "Part of skirt" , "Stitched edge of garment" , "Double-backed edge" , "try to attract attention" .)

I've seen this clue in the Evening Standard.
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