Cloth handbag at first could be ranked most important (12)
I believe the answer is:
handkerchief
'cloth' is the definition.
(handkerchief is a kind of cloth)
'handbag at first could be ranked most important' is the wordplay.
'at first' indicates taking the first letters.
'could be' indicates anagramming the letters.
'most important' becomes 'chief' (I've seen this in another clue).
The initial letter of 'handbag' is 'h'.
'ranked' anagrammed gives 'andker'.
'h'+'andker'+'chief'='HANDKERCHIEF'
(Other definitions for handkerchief that I've seen before include "Suit pocket accessory" , "Bandanna" , "Piece of material taken from Desdemona (Othello)" , "Piece of material used to wipe one's nose" , "Snot rag" .)