Obsessed when losing second glove (6)
I believe the answer is:
mitten
'glove' is the definition.
(I know that mitten is a type of glove)
'obsessed when losing second' is the wordplay.
'obsessed' becomes 'smitten' (smitten means obsessed with someone).
'when losing' indicates named letters should be taken away.
'second' becomes 's' (common abbreviation).
'smitten' with 's' removed is 'MITTEN'.
(Other definitions for mitten that I've seen before include "boxing glove" , "Hand-warmer" , "Garment" , "Hand warmer" , "Glove with no partition between fingers" .)