Language Beardsley uses occasionally (4)
I believe the answer is:
erse
'language' is the definition.
(historical term for Scottish Gaelic)
'beardsley uses occasionally' is the wordplay.
'uses occasionally' indicates alternate letters (letters taken at regular or occasional intervals).
The alternate letters of 'beardsley' are 'ERSE'.
(Other definitions for erse that I've seen before include "Old Celtic language" , "Possibly the language" , "Irish Gaelic language" , "Celtic tongue" , "Name sometimes given to languages of the Celts in Ireland and Scotland" .)