Shame about a bird on desert staying with new family? (9,5)

I believe the answer is:

paternity leave

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

'staying with new family?' is the definition.
(leave to spend time with a child)

'shame about a bird on desert' is the wordplay.
'shame' becomes 'pity' (synonyms).
'about' indicates putting letters inside.
'a bird on' becomes 'atern' (I am not sure about this - if you are sure you should believe this answer much more).
'desert' becomes 'leave' (desert can mean to leave or abandon).
'pity' placed around 'atern' is 'paternity'.
'paternity'+'leave'='PATERNITY LEAVE'

Can you help me to learn more?

(Other definitions for paternity leave that I've seen before include "time for Dad at home" , "Time off for new fathers" .)

I've seen this clue in The Times.

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