Starts to open some letters offering capital (4)
I believe the answer is:
oslo
'capital' is the definition.
(capital of Norway)
'starts to open some letters offering' is the wordplay.
'starts to' indicates taking the first letters.
The first letters of 'open some letters offering' is 'OSLO'.
(Other definitions for oslo that I've seen before include "where Ibsen lived under another name" , "Go solo to European capital" , "foreign port" , "City in Norway" , "Capital in throes of go-slow" .)