Laud new attempt to cover last of one's unfaithfulness (8)
I believe the answer is:
adultery
'unfaithfulness' is the definition.
(I've seen this before)
'laud new attempt to cover last of one's' is the wordplay.
'new' indicates an anagram.
'attempt to' becomes 'try' (I've seen this in other clues).
'cover' is an insertion indicator.
'last of' indicates one should take the final letters.
The final letter of 'one' is 'e'.
'laud' anagrammed gives 'adul'.
'adul'+'try'='adultry'
'adultry' placed around 'e' is 'ADULTERY'.
(Other definitions for adultery that I've seen before include "Betrayal of a partner" , "cheating" , "Moral lapse" , "form of betrayal" , "Act of infidelity to spouse" .)