In British English, 'surgery', meaning 'the treatment of injuries or diseases in people or animals by cutting open the body and removing
or repairing the damaged part' is a non-count (uncountable) noun. You
do not place either 'a' or 'the' before the word, and also it is not used in plural form ('surgeries') with this meaning. A single procedure is generally called an 'operation'
I had surgery on my knee following an accident
In older people, many hip operations take place because the joints may have been severely changed through osteoarthritis and may need to be replaced
In American and some other variants of English, 'surgery' can be used to mean the same thing uncountably, but the word may also be used countably (to mean a surgical operation or procedure), and in that case can be preceded with 'a' or 'the' and also be used plurally.
a doctor who has performed many surgeries
Get the arms you've always wanted with an arm surgery
In late December 2018, Mr. Houp was hospitalized at the VA Pittsburgh Hospital Center for a condition in his lower left leg and underwent a
leg surgery in January 2019
Some further confusion may arise because in countries using British English, 'surgery' is very often used as a countable noun to mean 'a location, room or building where a doctor (or dentist) sees patients and gives advice or treatment'.
Dr Jackson has her surgery at 226 Oaktree Road
In American English these places are usually called a doctor's or dentist's 'office'.