Little knowing, begins with a short period (4)
I believe the answer is:
week
'short period' is the definition.
'week' can be an answer for 'period' (week is a kind of period). I am not sure about the 'short' bit.
'little knowing begins' is the wordplay.
'little' becomes 'wee' (similar in meaning).
'begins' suggests taking the first letters.
The initial letter of 'knowing' is 'k'.
'wee'+'k'='WEEK'
'with a' is the link.
Can you help me to learn more?
(Other definitions for week that I've seen before include "Calendar period" , "A long time in politics?" , "Seven day period" , "Seven days" , "Part of the year" .)