'No way out' admits husband, carelessly (6)
I believe the answer is:
anyhow
'carelessly' is the definition.
(thesaurus)
'no way out admits husband' is the wordplay.
'out' is an anagram indicator (out can mean wrong or inaccurate).
'admits' indicates putting letters inside.
'husband' becomes 'h'.
'no'+'way'='noway'
'noway' with letters rearranged gives 'anyow'.
'anyow' enclosing 'h' is 'ANYHOW'.
(Other definitions for anyhow that I've seen before include "Randomly - at least" , "Be that as it may" , "In a muddle" , "In whatever case" , "In spite of what has been said" .)