Applaud those running across street in Derbyshire town (12)
I believe the answer is:
chesterfield
'derbyshire town' is the definition.
(town in Derbyshire)
'applaud those running across street' is the wordplay.
'applaud' becomes 'cheer' (cheering is a kind of applauding).
'those running' becomes 'field' (field can mean all the participants in a race).
'across' means one lot of letters goes inside another.
'street' becomes 'st' (abbreviation).
'cheer'+'field'='cheerfield'
'cheerfield' enclosing 'st' is 'CHESTERFIELD'.
'in' is the link.
(Other definitions for chesterfield that I've seen before include "Sofa; overcoat" , "Derby town?" , "Type of sofa - place in Derbyshire" , "Sofa; English town" , "somewhere to sit" .)