Hour goes back at the end of the day, disheartening midwife in Limerick, perhaps (5)
I believe the answer is:
rhyme
'limerick perhaps' is the definition.
(limerick is a kind of rhyme)
'hour goes back at the end of the day disheartening midwife' is the wordplay.
'hour' becomes 'hr' (abbreviation).
'goes back' says the letters should be written backwards.
'at the end of' suggests the final letters.
'disheartening' suggests removing the centre.
'midwife' with its centre removed is 'me'.
The final letter of 'day' is 'y'.
'hr' reversed gives 'rh'.
'rh'+'y'+'me'='RHYME'
'in' is the link.
(Other definitions for rhyme that I've seen before include "Short poem" , "Piece of poetry" , "often quoted with reason" , "Poem, eg for nursery" , "Sound correspondence" .)