Possessing and using a razor, not the first (6)

I believe the answer is:

having

'possessing' is the definition.
('have' can be a synonym of 'possess')

'using a razor not the first' is the wordplay.
'using a razor' becomes 'shaving' (eg shaving one's beard).
'not the first' means to remove the first letter.
'shaving' with its initial letter removed is 'HAVING'.

'and' is the link.

(Another definition for having that I've seen is " Possessing, owning".)

I've seen this clue in The Times.

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