Robin - I'm just a user, so others may have better tips, but, having experimented, the method below works for me (#):

1) Working in the cycle travel website, open the Winchester to Winchester route that you have created from the GPX file; add a via point on the existing route east of Etchilhampton; then click on that via point and select the option "Start here" (you will probably need to click on "more options" to reveal that menu item). For me, the KAW route now starts and finishes at the added via point ie it has rather magically joined the break in the circular route across Winchester.

2) place added via points on the KAW route a short distance either side of the one added at (1). (This will avoid the next action losing some of the KAW).

3) drag the via point added in (1) to your desired start point in Bath. (Assuming you have set the route type to be "Any", it will route from and to Bath via CycleRoute 4.)

You will now have a route that is from Bath, joining the KAW near Etchilhampton, then does a complete circuit of the KAW, then back to Bath. Note you can reverse the direction of the circuit if you click on the Reverse icon shown in the Route Tools. Don't foget to save!

Ironically, I think the GPX file you are starting from (the Cycle UK website?) was actually done for them by Richard, using Cycle Travel ...

 (# I would probably ignore the GPX and instead simply build the route from scratch in Cycle Travel, using via points to build the route following KAW, breaking it up into daily sections...) 

 I will be interested to see it RF or others have better methods, but meantime, HTH.

Robin, answering your question would be easier if you had followed the advice in the red box above your initial post.

The first thing to be aware of is the help page. admittedly not the easiest to find.

Since you haven't shared the gpx you are using I am using this one, created by a user @BradHelper and using the website, not the app. 

What I would do is add a viapoint as close as possible to the start point and one as close as possible to the end point. Then, I would carefully drag the start point to Bath. (It may need to be done in stages). Once located, I'd do the same with the end point. Voila! It takes seconds. 

I can 'play around' with the Bath to Winchester legs to vary the route. 

However, personally speaking, that doesn't exactly look like the most efficient route!

As far as I know, adding 'new' start and end points is going to be tricky, especially on such a complicated (and off road) route. So, me being lazy, I'd plan a more efficient (and less 'doubling back') route. 

@Simon Day I do not see the option to 'Start here' on the route I am using, possibly because it is not a circular route.

To do that, I'd download the BradHelper gpx file, start from scratch and display that gpx as a 'Trace'. Then I'd start in Bath, and plot a route to the most practical place to join the 'Trace'. Using the 'Trace' I'd create a route along the KAW and back to home. (Trace is available under the 'Hamburger' menu, top right of the map screen)

Note: Meaning no disrespect to @BradHelper, I am not a fan of using random gpx files from the internet. We don't know when they were made, the attitude of the creator to 'rough stuff', ascent/descent, the season they were planned for or even the time of day! In other words, I suggest other people's routes be treated as a guideline as opposed to a blueprint. 

Of course, having written all that, the easiest is simply to leave the KAW route as it is and simply plot a second route to join it wherever you want. By saving both of those in the same folder you'll see the 'entire' journey on the one map as two (possibly 3 - leg home) individual routes. App, gps or PDF navigation is barely more complicated. 

@Hobbes, Re "Start from here"

In my experimenting I first used the search function in CT (the "magnifying glass" symbol at the top right of the home page(s) to seek "King Alfred's Way"; that produces links to a Route Guide (which is a broken page), and to various user-created routes all with the same name. 

The first of the user-created routes is Reading to Reading, by Any surface. Note like Robin's GPX-source route, it is not a true circular, in that the start any finish are on north and south side of Reading railway station. When I click on any via point in that route (an existing point or an added one) the "start here" menu option exists and behaves in the manner I described. 

I have replicated this by creating my own near clone version of Reading  (north side) to Reading (south side) via a rough version of KAW; the "start here" option is available and behaves as above. (Saved just now as a public route named "Test for start here" https://cycle.travel/map/journey/914187) 

The odd thing is that the "start here" option does not always appear in routes ...eg the 6th user-created route returned by my search for KAW is a version from Winchester to a slightly different point in Winchester; so seemingly conceptually like the Reading example and my test, but the "start here option does not come up. Seems one where we need Richard's guidance on what is happening?

From further experimenting: it seems the "start here" menu option always appears when the start and finish of a "horseshoe" route are within a few hundred metres of each other - seemingly up to approx 200m apart; otherwise it is not offered. It does not appear to be affected by the length of the route (I had wondered if it might be proportional to the route length, but seemingly not).

So it does offer a universal way of editing "very nearly cicular" routes to make them start/finish at a different point on the route; and converts them into true circulars as well.

@ Robin: your GPX-sourced route might need a tweak to bring the start and end points in Winchester close enough together to reveal this menu feature.

(All this only in the website version, of course.)