Unctuous server in run-down café (6,5)

I believe the answer is:

greasy spoon

I'm a little stuck... Click here to teach me more about this clue!

'run-down cafe' is the definition.
'greasy spoon' can be an answer for 'cafe' (I've seen this before). I'm not certain of the 'run-down' bit.

'unctuous server' is the wordplay.
'unctuous' becomes 'greasy' (I've seen this before).
'server' becomes 'spoon'.
'greasy'+'spoon'='GREASY SPOON'

'in' is the link.

Can you help me to learn more?

(Other definitions for greasy spoon that I've seen before include "diner, by the way" , "Naff caff" , "Roadside cafe" , "cheap place to eat" , "Workers' cafe" .)

I've seen this clue in The Irish Times.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.