American finishes major public search on foot (4)
I believe the answer is:
arch
'on foot' is the definition.
(I've seen this before)
'american finishes major public search' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'finishes' suggests the final letters (I've seen 'finish of' mean this).
The final letters of 'major public search' are 'rch'.
'a'+'rch'='ARCH'
(Other definitions for arch that I've seen before include "Masonry structure for spanning an opening" , "Curved entrance" , "Curved aperture" , "Curved span" , "Shrewd; bridge" .)