Crossword Genius

Carry on and have a pet songbird (4,2,2)

Ross

I believe the answer is:

keep at it

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

'carry on' is the definition.
(I've seen this before)

'have a pet songbird' is the wordplay.
I cannot quite see how this works, but
'have' could be 'keep' (keeping is a kind of having) and 'keep' is present in the answer.
'a' is found in the answer.
'songbird' could be 'tit' (tit is a type of songbird) and 'tit' is found in the answer.
This accounts for all the letters.
This explanation may well be incorrect...

'and' acts as a link.

Can you help me to learn more?

(Other definitions for keep at it that I've seen before include "Continue" , "Persevere" , "carry on" .)

I've seen this clue in the Sydney Morning Herald.
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