Kind to support military’s first exploit (4)
I believe the answer is:
milk
'exploit' is the definition.
(to milk something is to exploit its potential)
'kind to support military's first' is the wordplay.
'kind' becomes 'ilk' (ilk is a kind of kind).
'to support' means one lot of letters go next to another (in a down clue, some letters hold others up).
'military's first' becomes 'm' (1st letter of 'military').
'ilk' put after 'm' is 'MILK'.
(Other definitions for milk that I've seen before include "Extract money from" , "an early delivery?" , "Nourishing fluid produced by mammals" , "Dairy product" , "Exploit" .)