Best way to see Cambridge, York, Canterbury/Leeds

I'm not quite sure if you mean 4 days in London and then those trips, or if it is all to be squeezed into the 4 days. If the latter, while you could do this with a combination of train and bus, in 4 days you'd be spending most of your time travelling. I'd suggest train to York then hire a car and come south via Cambridge and skirt London for Kent. The disadvantage would be that this would be a one-way hire and that tends to be more pricey. If you'd prefer to stick to public transport, cut something out.

For Canterbury - use the train.

Another trip = train from London (Kings Cross) to Cambridge - see Cambridge and then train to York (you will need to change at Peterborough). It may work out cheaper to split the train ticket at Peterborough rather than buy a through ticket Cambridge > York.
www.nationalrail.co.uk

For travel between London (St Pancras and Victoria) and Canterbury (East and West) try going to the website for Southeastern Railway -- https://www.southeasternrailway.co.uk/

National Rail is a portal to re-direct you to the correct railway company site. It does not offer different tickets.
What you want is a "day return" to Canterbury.. As the name suggests, it allows you to come back on any train on the same day. Open returns, which are for coming back on a different day, are normally not available for short trips, like London to Cantrerbury..

Your title confused me. ".... Canterbury/Leeds". Leeds is a large city (¾ million, 4th largest city in the country) in northern England. Leeds castle is near the small village of Leeds, Kent (population 790) and does not have a railway station. Do not try and look up trains to "Leeds" or you will end up in a large metropolis at the wrong end of the country

Sorry, meant Leeds Castle. Thanks all for excellent feedback.

Here's the link I bookmarked for myself, for visiting Leeds Castle.

https://www.leeds-castle.com/Visit/Getting+to+Leeds+Castle