Little Princess's beginning to cry (4)
I believe the answer is:
weep
'cry' is the definition.
('weep' can be a synonym of 'cry')
'little princess's beginning' is the wordplay.
'little' becomes 'wee' ('wee' can be similar in meaning to 'little').
'beginning' indicates taking the first letters.
The first letter of 'princesss' is 'p'.
'wee'+'p'='WEEP'
'to' is the link.
(Other definitions for weep that I've seen before include "Seep slowly out" , "Bawl" , "Sob - ooze" , "Display sorrow" , "Greet" .)