Second egg on tray -- not oddly something one gets in hospital? (7)

I believe the answer is:

surgery

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'something one gets in hospital?' is the definition.
I can't tell whether this definition defines the answer.

'second egg on tray not oddly' is the wordplay.
'second' becomes 's' (common abbreviation).
'egg on' becomes 'urge' (to egg on is to urge or encourage).
'not oddly' indicates alternate letters (odd letters are removed).
The alternating letters of 'tray' are 'ry'.
's'+'urge'+'ry'='SURGERY'

Can you help me to learn more?

(Other definitions for surgery that I've seen before include "NHS cuts" , "Medical intervention" , "Doctors' practice" , "operation" , "Doctor's premises" .)

I've seen this clue in The Telegraph.

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