Swear that cures are possible (5)

I believe the answer is:

curse

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

'swear' is the definition.
('curse' can be a synonym of 'swear')

'cures are possible' is the wordplay.
'cures' anagrams to 'CURSE'.
But, I am unsure how this is indicated.

'that' acts as a link.

Can you help me to learn more?

(Other definitions for curse that I've seen before include "Imprecation or oath" , "Blasphemy" , "Blaspheme; damn" , "Invocation of evil; swear" , "Cuss" .)

I've seen this clue in the Evening Standard.

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