Embarrassed, Brook to bore another hole (7)

'embarrassed brook' is the definition.
I know nothing about this answer so I can't tell whether this works.

'bore another hole' is the wordplay.
I cannot quite see how this works, but
'bore' could be 'drill' (I've seen this in another clue) and 'drill' is found in the answer.
'another' could be 're' (I've seen this in other clues) and 're' is located in the answer.
No letters remain.
This may be the basis of clue (or it may be nonsense).

'to' is the link.

Can you help me to learn more?