Suffer from sunburn in exercise the French offered up (4)
I believe the answer is:
peel
'suffer from sunburn' is the definition.
(I've seen this in another clue)
'exercise the french offered up' is the wordplay.
'exercise' becomes 'pe' (Physical Education).
'the french' becomes 'le' ('the' in French).
'offered up' says the letters should be written backwards.
'le' back-to-front is 'el'.
'pe'+'el'='PEEL'
'in' acts as a link.
(Other definitions for peel that I've seen before include "Remove the rind" , "old prime minister" , "Lemon skin" , "5,27 [PRIME MINISTER]" , "Citrus skin" .)