Pompous leader has tattoo after first month on vessel (10)
I believe the answer is:
panjandrum
'pompous leader' is the definition.
(a person who claims authority)
'tattoo after first month on vessel' is the wordplay.
'tattoo' becomes 'drum'.
'after' means one lot of letters go next to another.
'first month' becomes 'jan' (abbreviation for January).
'on' means one lot of letters go next to another.
'vessel' becomes 'pan' (vessel or container for cooking).
'drum' put after 'jan' is 'jandrum'.
'jandrum' put after 'pan' is 'PANJANDRUM'.
'has' acts as a link.
(Other definitions for panjandrum that I've seen before include "Big cheese" , "man of great influence" , "influential man" , "pompous official" , "Person in high authority" .)