The wren flew inside, dear, as one got close to it (4,4)

I believe the answer is:

drew near

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'one got close to it' is the definition.
I can't tell whether this definition defines the answer.

'the wren flew inside dear' is the wordplay.
'flew' is an anagram indicator (I've seen 'fly' mean this).
'inside' means one lot of letters goes inside another.
'wren' anagrammed gives 'rewn'.
'rewn' going into 'dear' is 'DREW NEAR'.

'as' acts as a link.

Can you help me to learn more?

I've seen this clue in the Evening Standard.

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