Crossword Genius

Saint with primate seen around church area (4)

Ross

I believe the answer is:

apse

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

'church area' is the definition.
(I've seen this before)

'saint with primate seen' is the wordplay.
I cannot really understand how this works, but
'saint' could be 's' and 's' is located in the answer.
'primate' could be 'ape' (ape is a kind of primate) and 'ape' is present in the leftover letters.
No letters remain.
This explanation may well be incorrect...

'around' is the link.

Can you help me to learn more?

(Other definitions for apse that I've seen before include "Rounded end of building" , "may be seen in church!" , "Recess with domed rof in a church" , "room for an altar" , "Church recess with domed or arched roof" .)

I've seen this clue in The Times.
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