To let a thousand in can cause distress (7)
I believe the answer is:
torment
'distress' is the definition.
(I know that torment is a type of distress)
'to let a thousand in' is the wordplay.
'let' becomes 'rent' (synonyms).
'a thousand' becomes 'M' (Roman numeral for a thousand).
'in' indicates putting letters inside.
'rent' placed around 'm' is 'rment'.
'to'+'rment'='TORMENT'
'can cause' is the link.
(Other definitions for torment that I've seen before include "Persecute" , "Severe physical or mental suffering" , "Punish" , "Pester annoyingly" , "Inflict pain on - pester" .)