Eliminate ibises, initially, by heartless culling and mocking (10)

'mocking' is the definition.
(ridiculing is a kind of mocking)

'eliminate ibises initially by heartless culling' is the wordplay.
'eliminate' becomes 'rid' (I've seen this in another clue).
'initially' indicates taking the first letters.
'by' means one lot of letters go next to another.
'heartless' means to remove the middle letters.
'culling' with its middle removed is 'culing'.
The initial letter of 'ibises' is 'i'.
'rid'+'i'+'culing'='RIDICULING'

'and' is the link.