Casually read lines penned in British English (6)
I believe the answer is:
browse
'casually read' is the definition.
(to browse is to read in a casual way)
'lines penned in british english' is the wordplay.
'lines' becomes 'rows' (row is a kind of line).
'penned in' indicates putting letters inside.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'english' becomes 'e' (abbreviation).
'b'+'e'='be'
'rows' placed inside 'be' is 'BROWSE'.
(Other definitions for browse that I've seen before include "Graze - glance through a book, magazine etc" , "Look at sites 13 ac" , "Eat by continually nibbling" , "Look through in a random way, a bookshop perhaps" , "(Of animal) feed on leaves, twigs, etc." .)