Saunter extremely meekly around old Kent area (5)
I believe the answer is:
mosey
'saunter' is the definition.
(both can mean to walk in a leisurely manner)
'extremely meekly around old kent area' is the wordplay.
'extremely' says to hollow out the word (remove centre letters) (extreme, outside letters only).
'around' indicates putting letters inside.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'kent area' becomes 'se'.
'meekly' with its centre taken out is 'my'.
'o'+'se'='ose'
'my' going around 'ose' is 'MOSEY'.
(Another definition for mosey that I've seen is " Make off".)