Hi! Physics BS, but they let me take some grad courses, including a Spacetime and Relativity class. I can help.
The word "mass" is used in physics in three different general contexts. First, we have mass in mass-energy, as in "how much energy can I get for trading in this mass?" Mass-energy is the coin paid as the price of existence. If it exists, it has mass-energy. Mostly mass for us. Mostly. We can skip that one for now.
The second context of mass is inertial. Mass has the property of inertia, of resisting a change in its direction or speed. It resists stopping if it is motion, and if it is stopped, it resists moving. The degree of the resistance is also called mass. Put a pin in this one.
The third context of mass is gravitational. Two masses, attracting one another because a force between them, a force which is not based on charge or the relatively nearby exchange of some more exotic bosons, no, just attraction based on how much mass is present. Nothing more special.
Now, curiously, values of each one of these seem to agree!
Einstein's absolute core concept in general relativity, the idea from which all else is built, is that inertial mass is identical to gravitational mass, not merely in number, but so fundamentally intertwined that there is no real difference between them, other than being two faces of the same coin. Now, that does not sound like much, but it gives birth to experiments such as an elevator which is falling toward versus an elevator floating far from gravitational sources, and that they are, from the inside of the elevator, impossible to differentiate.
Einstein then constructs general relativity from this, that the "m" in "F = ma" is identical to the first m in "F = -G m1 * m2 / r^2"
In MOND, the two ms are not identical, they only appear close most places, and so you cannot construct general relativity atop it. You will get most correct approximations but you're missing out in some cases.