Thickness of hose from end — fire's beginning to go out (6)
I believe the answer is:
denier
'thickness of hose' is the definition.
(unit used to measure the thickness of hosiery)
'end fire's beginning to go out' is the wordplay.
'beginning to go' means to remove the first letter.
'out' indicates anagramming the letters (out can mean wrong or inaccurate).
'fire' with its initial letter taken off is 'ire'.
'end'+'ire'='endire'
'endire' anagrammed gives 'DENIER'.
'from' acts as a link.
(Other definitions for denier that I've seen before include "Unit of weight for fine material, silk or nylon" , "Weight unit by which fineness of yarn is measured" , "One who repudiates" , "One who doesn't believe" , "Peter?" .)