Hardship in Cornish town not unknown (7)
I believe the answer is:
penance
'hardship' is the definition.
(I've seen this before)
'cornish town not unknown' is the wordplay.
'cornish town' becomes 'penzance' (I've seen this before).
'not' indicates named letters should be taken away.
'unknown' becomes 'z' (algebra).
'penzance' with 'z' taken out is 'PENANCE'.
'in' is the link.
(Other definitions for penance that I've seen before include "Voluntary self-punishment" , "Voluntary self-abasement" , "Act of penitence" , "Punishment inflicted on oneself" , "Expression of sorrow for sin" .)