I have bought an advance ticket from Rye (Sussex) to London St Pancras. However, for various reasons I'm now thinking of hiring a car in Ashford and would obviously need to return it there as well. Will I be able to use my ticket to get through the barriers at Ashford and join the train I'm booked on (since the journey I'm booked on changes trains at Ashford anyway)?
Break of journey, such as starting late, is never allowed on an Advance ticket. However, if you can get through the Ashford barriers - for example with a cheap local single such as Ashford to Ham Street, £3.40 Off-Peak Day Single/£3.70 Anytime (I very much doubt your Advance will work the barriers and I certainly wouldn't try) - then it is unlikely your break of journey will be discovered, unless you have the extreme misfortune of the same conductor checking all tickets on both booked trains (and noticing you weren't there on the previous train).
Whilst I would never encourage breaking one's journey where you know it is not permitted to do so, in most situations the worst penalty for doing if you were discovered would be to have to pay the difference between your Advance and the cheapest walk-up fare that would have let you break your journey.
Alternatively, you could regularise the situation now by excessing your Advance to the cheapest relevant walkup ticket (as I believe virtually none of the restriction codes in that area ban break of journey). It may, of course, simply be cheaper to get a new ticket from Ashford to London, depending on how much you paid for your Advance and when you are travelling.