Rude song about hotel cat (9)

'cat' is the definition.
(shorthair is a type of cat)

'rude song about hotel' is the wordplay.
I cannot really understand how this works, but
'rude' could be 'short' (short can mean rude or abrupt) and 'short' is located in the answer.
'song' could be 'air' (I've seen this before) and 'air' is present in the answer.
'hotel' could be 'h' (phonetic alphabet: alpha, bravo, charlie etc.) and 'h' is located in the answer.
No letters remain.
This explanation may well be incorrect...

Can you help me to learn more?