American gym teacher perhaps a distant relative? (3-3)
I believe the answer is:
ape-man
'a distant relative?' is the definition.
'american gym teacher perhaps' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'gym teacher perhaps' becomes 'PE man' (physical education man).
'a'+'peman'='APE-MAN'
(Other definitions for ape-man that I've seen before include "Tarzan, for example" , "One of our extinct ancestors?" , "A not-quite-human ancestor" , "Extinct early primate" , "Long extinct human ancestor" .)