People following old lady in bleak, poor state (6,8)
I believe the answer is:
banana republic
'poor state' is the definition.
'people following old lady in bleak' is the wordplay.
'people' becomes 'public' (**).
'following' means one lot of letters go next to another.
'old lady' becomes 'nana'.
'in' means one lot of letters goes inside another.
'bleak' becomes 'bare' ('bare' can be a synonym of 'bleak').
'nana' inserted within 'bare' is 'bananare'.
'public' put after 'bananare' is 'BANANA REPUBLIC'.
(Other definitions for banana republic that I've seen before include "weak state" , "Small politically unstable country dominated by foreign companies" , "state precariously run" , "Small state dependant on one export" .)