Tim wrote:

I have a zone 1-3 travelcard loaded on my Oyster card, and want to travel
from Clapham Junction to Kew Gardens. This involves getting a train to
Richmond then changing for Kew Gardens.

The 337 bus goes frequently from right outside Clapham Junction (leave
through the shopping centre, then cross the road) to Richmond town centre.
It takes about 40 minutes in weekday traffic. Then 65/391 to Kew as others
have said.

Since the 337 goes past East Putney station, you could take the District up
to Earls Court and down to Kew if you really wanted.

Does anyone know whether I can (legaly) travel through Richmond without
buying an extension, since my starting point & destination are bot covered
by my travelcard (and I don't pass any ticket gates, just walk from one
platform to another)?

You would pass a gateline, in the corridor connecting platform 1 (the
westbound SWT platform) to the rest of the station.

Alex