Little person's beginning to cry (4)
I believe the answer is:
weep
'to cry' is the definition.
(synonyms)
'little person's beginning' is the wordplay.
'little' becomes 'wee' (similar in meaning).
'beginning' suggests taking the first letters.
The initial letter of 'person' is 'p'.
'wee'+'p'='WEEP'
(Other definitions for weep that I've seen before include "lament" , "Keen" , "Turn on the waterworks" , "pipe" , "Display sorrow" .)