The longing teachers have at heart (4)

I believe the answer is:

ache

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'heart' is the definition.
Although both the answer and definition are singular nouns, I can't understand how one could define the other.

'the longing teachers have' is the wordplay.
'have' indicates a hidden word.
'ACHE' can be found hidden inside 'longing teachers'.

'at' is the link.

Can you help me to learn more?

(Other definitions for ache that I've seen before include "Dull throbbing pain" , "Similar" , "Long ? pain" , "Persistent dull pain" , "Hanker" .)

I've seen this clue in the Evening Standard.

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