Crossword Genius

Sort of cat a chap gets cross with? (4)

Ross

I believe the answer is:

manx

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'sort of cat' is the definition.
Although both the answer and definition are singular nouns, I can't understand how they can define each other.

'a chap gets cross with?' is the wordplay.
'a chap' becomes 'man' (I've seen this before).
'gets' is a charade indicator (letters next to each other).
'cross with?' becomes 'x' (I am not sure about the 'with?' bit.**).
'man'+'x'='MANX'

Can you help me to learn more?

(Other definitions for manx that I've seen before include "Sort of cat (from Douglas?)" , "Kind of cat" , "The natives' language" , "From an isle in the Irish Sea" , "To do with the Isle of Man - say a cat" .)

I've seen this clue in the Evening Standard.
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