One eliminating small old city in type of game (9)
I believe the answer is:
destroyer
'one eliminating' is the definition.
The answer is a person as well as being a singular noun. This is suggested by the definition.
'small old city in type of game' is the wordplay.
'small' becomes 's' (abbreviation - e.g. clothes size).
'old city' becomes 'Troy' (ancient city).
'in' is an insertion indicator.
'type of game' becomes 'deer' (I am not sure about this - if you are sure you should believe this answer much more).
's'+'troy'='stroy'
'stroy' inserted inside 'deer' is 'DESTROYER'.
Can you help me to learn more?
(Other definitions for destroyer that I've seen before include "heavily armed warship" , "One scuppers" , "Large warship" , "Military vessel" , "Small, fast warship" .)