What Bragg meant by now, I cannot know. There is a huge amount of writers who meant all sort of things, and it's impossible to know all these views.
The question whether the quantum system (quantum particle) is a wave or is a particle, preoccupies the physicists even today, 91 years after de Broglie formulated his $\lambda = h/p$ formula.
You see, both concepts, particle and wave, are classical. And the quantum mechanics loughs at us in our face of our classical concepts. So, I'll tell you what is believed today, a few approaches:
A) (Standard Quantum Mechanics) The quantum systems are wave-like, i.e. pass simultaneous through two slits, produce interference. However, in the macroscopic measurement apparatuses, they undergo decoherence, then collapse, and on a photographic plate the wave delivers all its energy to a single molecule.
In short, we know nothing more than the wave-function tells us, we don't know of the existence of a particle. The particle is some classical idea of us, because we are eluded by the discrete reaction of the measurement apparatus (the collapse) - the single point on the photographic plate.
B) (Bohm's interpretation) The quantum particle is almost a classical particle, but it floats in a quantum potential produced by the wave-function, and this potential guides the particle about where it is allowed to be and where not.
But this interpretation has problems with the relativity, in short, one has to assume that for the quantum system there exists a preferred frame of coordinates - idea refuted by the relativism.
C) (Asher Peres) We have to change our conceptions of space-time, so as to be able to understand entanglements and the quantum systems in general.
But Peres didn't have time to say something more clear, he passed away.
D) GRW (Ghirardi-Rimini and Weber) interpretation, that says that we have to add to the Schrodinger equation a stochastic term and a nonlocal potential, and that modification will produce for each trial in an experiment, a well-defined result. Though, this interpretation has also some flaws.
E) We have a classical mind and therefore, no chance to understand the quantum behavior. Therefore, we'd better take the wave-function and shut-up and calculate probabilities.
Needless to say, I don't subscribe to this, I agree with Peres.
Next, you ask : "Does the matter wave carry energy & mass? Previously I thought they are just probability waves. But a statement from Resnick & Halliday : " We should expect wavefunction to be more complicated than the corresponding quantity for a light wave as a matter wave in addition to energy & momentum, transports mass & (often) electric charge" . So, is it true?"
As I said above, there are so many views! In the Standard QM there is no particle, we have only the sort of wave, and at each point and point inside the wave there appear all the point-like properties, mass, charge. About energy, if the wave-function is a superposition of energy eigenstates, then the energy and momentum are undefined, so it's a bit difficult to answer Reasnik & Halliday exactly. What yes is defined is their average. But indeed the matter-wave carries energy and momentum.
By Bohm's interpretation, the particle has no other properties than impress a detector. (How so, without properties, it's not clear.) All the properties belong to the quantum potential.
