I just submitted my third paper to the arXive, you can find it here. This completes my DPhil project by proving that Cayley fibrations are stable under small perturbations of the Spin(7)-structure. This is done by looking at how nearly singular Cayleys behave under perturbation. It turns out that at least up to 
they behave as expected, with the gluing described in my second paper giving a good approximation. We do this by solving the linear Cayley equation on the glued manifold and showing that the solutions (which describe the first derivative of the Cayley fibration so to speak) can also be obtained by gluing well-understood pieces. We use this to complete the longstanding programme of Kovalev to construct coassociative fibration on twisted connected sum 
manifolds.