MOND has nothing to say about the curvature of spacetime, since MOND is Newtonian (MOdified Newtonian Dynamics). It goes back to "F=ma and gravity is a force" and modifies the rules so that gravity grows weaker faster at a certain scale.

The fact that MOND fits a lot of the data troubled cosmologists, because they know that a General Relativistic theory is needed to explain pretty much the rest of gravity.

TeVeS is an extension to General Relativity that reduces to MOND in the non-relativistic limit. For comparison, General Relativity reduces to Newtonian gravity in the non-relativistic limit. The non-relativistic limit is when speeds and spacetime curvature are small.