Burrower crosses river, primarily heading for animal shelter (9)
I believe the answer is:
mousehole
'animal shelter' is the definition.
(shelter used by a mouse)
'burrower crosses river primarily heading' is the wordplay.
'burrower' becomes 'mole' (I've seen this in another clue).
'crosses' indicates putting letters inside.
'river' becomes 'ouse' (Yorkshire river).
'primarily' suggests taking the first letters.
The initial letter of 'heading' is 'h'.
'ouse'+'h'='ouseh'
'mole' placed around 'ouseh' is 'MOUSEHOLE'.
'for' is the link.
(Other definitions for mousehole that I've seen before include "in Cornwall" , "Little creature's home" , "Opening made by rodent" , "Murine retreat" .)