Frenchman's gym slip one's put on splits (6)
I believe the answer is:
pierre
'frenchman's' is the definition.
(common French name)
'gym slip one's put on splits' is the wordplay.
'gym' becomes 'pe' (short for Physical Education).
'slip' becomes 'err' (to err is to make an error).
'one' becomes 'i' (Roman numeral).
'put on' means one lot of letters go next to another.
'splits' means one lot of letters goes inside another (inserted letters split open the word).
'err' after 'i' is 'ierr'.
'pe' placed around 'ierr' is 'PIERRE'.
(Other definitions for pierre that I've seen before include "See 17" , "French boy's name" , "French first name" , "Frenchman" , "Peter in France" .)