Unending plea to get Rex to emerge (6)
I believe the answer is:
appear
'to emerge' is the definition.
(I've seen this in another clue)
'unending plea to get rex' is the wordplay.
'unending' means to remove the last letter (the word doesn't have its end letter).
'plea' becomes 'appeal' (I've seen this in another clue).
'to get' says to put letters next to each other.
'rex' becomes 'R' (abbreviation e.g. GR=George Rex, King George).
'appeal' with its last letter removed is 'appea'.
'appea'+'r'='APPEAR'
(Other definitions for appear that I've seen before include "Look - materialise" , "Seem or materialise" , "Make an entrance as ghost may" , "hove in sight" , "Present oneself" .)