Be sure Ireland will shelter no undesirable (4,5)

I believe the answer is:

bete noire

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

'be sure ireland will shelter no' is the wordplay.
'be sure' becomes 'bet' ('I bet that...' means 'I am sure that...').
'ireland will' becomes 'eire' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'shelter' means one lot of letters goes inside another.
'eire' going around 'no' is 'enoire'.
'bet'+'enoire'='BETE NOIRE'

Can you help me to learn more?

(Other definitions for bete noire that I've seen before include "One's particular bugbear" , "Pet hate" , "Intensely disliked thing" , "One's particular hate" , "Person or thing one particularly dislikes" .)

I've seen this clue in The Independent.

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