Small building suffers, having no entry places for pets (7)
I believe the answer is:
hutches
'places for pets' is the definition.
'small building suffers having no entry' is the wordplay.
'small building' becomes 'hut'.
'suffers' becomes 'aches' ('ache' can be a synonym of 'suffer').
'having no entry' means to remove the first letter.
'aches' with its first letter taken off is 'ches'.
'hut'+'ches'='HUTCHES'
(Other definitions for hutches that I've seen before include "Wooden cages for small animals" , "Cages for small animals, rabbits say" , "Small boxes for rabbits and other small animals" .)