Desparately poor, saint Anthony shortly caved in (10)

I believe the answer is:

stonybroke

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'saint anthony shortly caved in' is the definition.
I don't know anything about this answer so I can't judge whether it can be defined by this definition.

'desparately poor' is the wordplay.
'desparately' becomes 'stony' (I can't justify this - if you can you should give a lot more credence to this answer).
'poor' becomes 'broke' (similar in meaning).
'stony'+'broke'='STONYBROKE'

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(Other definitions for stonybroke that I've seen before include "on the rocks" , "penniless" .)

I've seen this clue in The Irish Times.

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