During half day's closing, ran back for school (8)
I believe the answer is:
seminary
'for school' is the definition.
(seminary is a kind of school)
'during half day's closing ran back' is the wordplay.
'during' indicates putting letters inside.
'half' becomes 'semi' (I've seen this before).
'closing' suggests the final letters (the letter which ends or closes the word).
'back' says the letters should be written in reverse.
The final letter of 'day' is 'y'.
'ran' in reverse letter order is 'nar'.
'semi'+'y'='semiy'
'semiy' going around 'nar' is 'SEMINARY'.
(Other definitions for seminary that I've seen before include "private school" , "College for trainee priests" , "Where ministers could have learnt" , "Theological college" , "Theological school" .)