Struggle to hold cross new reindeer (5)
I believe the answer is:
vixen
'reindeer' is the definition.
(Vixen is one of Santa's reindeer)
'struggle to hold cross new' is the wordplay.
'struggle' becomes 'vie' (I've seen this before).
'to hold' indicates putting letters inside.
'cross' becomes 'x' (X is cross-shaped).
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'vie' placed around 'x' is 'vixe'.
'vixe'+'n'='VIXEN'
(Other definitions for vixen that I've seen before include "Female fox - spiteful woman" , "Female with a brush" , "A foxy lady?" , "Bad-tempered woman - earth mother?" , "She-fox" .)