Little Boy's younger and nastier cousin admitting personal problem to hospital doctor (1-4)
I believe the answer is:
h-bomb
'little boy's younger and nastier cousin' is the definition.
(Little Boy was a type of atomic bomb)
'admitting personal problem to hospital doctor' is the wordplay.
'admitting' indicates putting letters inside (admit can mean to welcome in or accept).
'personal problem' becomes 'bo' (informal abbreviation for body odour).
'to hospital' becomes 'H' (abbreviation for hospital on maps).
'doctor' becomes 'MB' (Bachelor of Medicine).
'h'+'mb'='hmb'
'bo' put within 'hmb' is 'H-BOMB'.
(Other definitions for h-bomb that I've seen before include "Hydrogen weapon" , "Thermonuclear weapon" , "Extremely powerful weapon" , "Atomic weapon" , "powerful arm" .)