Palm beginning to sway in the past (4)
I believe the answer is:
sago
'palm' is the definition.
(sago palm is a kind of palm)
'beginning to sway in the past' is the wordplay.
'beginning to' indicates taking the first letters.
'in the past' becomes 'ago' ('ago' can be a synonym of 'in the past').
The initial letter of 'sway' is 's'.
's'+'ago'='SAGO'
(Other definitions for sago that I've seen before include "Powdery starch used in cooking" , "Powdery starchy food" , "Edible powdery starch" , "Starchy foodstuff" , "Palm cereal" .)