Browse and look round central part of library (5)
I believe the answer is:
graze
'browse' is the definition.
('graze' can be a synonym of 'browse')
'look round central part of library' is the wordplay.
'look' becomes 'gaze' (gaze is a kind of look**).
'round' means one lot of letters goes inside another.
'central part of' means to look at the middle letters.
The centre of 'library' is 'r'.
'gaze' going around 'r' is 'GRAZE'.
'and' acts as a link.
(Other definitions for graze that I've seen before include "Scrape - feed on grass" , "Eat lightly and often" , "Browse; scratch" , "Feed in meadow" , "Brush" .)