Feeling Irish Republic must shelter no undesirable (4,5)
I believe the answer is:
bete noire
'undesirable' is the definition.
(I've seen this before)
'feeling irish republic must shelter no' is the wordplay.
I cannot quite see how this works, but
'feeling' could be 'ire' (ire is a kind of feeling) and 'ire' is present in the answer.
'no' could be 'o' (resembles 0 - 'no' can mean 'zero') and 'o' is present in the answer.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for bete noire that I've seen before include "One particularly disliked" , "One's particular bugbear" , "Intensely disliked thing" , "Particular dislike" , "Object of aversion" .)