Using small shack, I reportedly sleep (7)

I believe the answer is:

shuteye

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'sleep' is the definition.
(I know that shuteye is a type of sleep)

'using small shack i reportedly' is the wordplay.
'using small' becomes 's' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'shack' becomes 'hut' (synonyms).
'reportedly' indicates a 'sounds like' (homophone) clue.
'i' sounds like 'eye'.
's'+'hut'+'eye'='SHUTEYE'

Can you help me to learn more?

(Other definitions for shuteye that I've seen before include "Slangy sleep with closed optic" , "zeds" , "nap" , "Slang sleep, literally described" , "Sleep (colloq.)" .)

I've seen this clue in The Daily Express.

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