Extend short programme covering half of our capital (7)
I believe the answer is:
prolong
'extend' is the definition.
(I know that extend can be written as prolong)
'programme covering half of our capital' is the wordplay.
'programme' becomes 'prog' (short for programme).
'covering' is an insertion indicator.
'half of' means to take half the letters.
'our capital' becomes 'London' (capital of the UK).
'london' halved is 'lon'.
'prog' enclosing 'lon' is 'PROLONG'.
'short' acts as a link.
(Other definitions for prolong that I've seen before include "Stretch out" , "Extend, protract" , "Extend the duration of" , "Extend in length" , "Draw out" .)