Swallow might sing with swift, finally, at the end of June. (6)
I believe the answer is:
ingest
'swallow' is the definition.
(I've seen this before)
'might sing with swift finally at the end of june' is the wordplay.
'might' is an anagram indicator (I've seen this in other clues).
'with' means one lot of letters go next to another.
'finally' says to take the final letters.
'at the end of' suggests the final letters.
The final letter of 'swift' is 't'.
The last letter of 'june' is 'e'.
't'+'e'='te'
'sing' put after 'te' is 'tesing'.
'tesing' anagrammed gives 'INGEST'.
(Other definitions for ingest that I've seen before include "Scoff" , "Absorb, assimilate" , "take in, eat" , "Eat - gin set (anag)" , "Take in with tinges" .)